563:
1622:
4075:
1032:. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
1711:
1321:
8144:
979:
1753:
1150:
2020:
587:
549:
4338:
2262:
45:
4172:". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).
1871:. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered
3910:
3607:. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.
6089:, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.
3661:
4790:
3552:
but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is
4271:
radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510°
4897:
5189:
1591:
340:
might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see
2968:
equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One
1173:, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called
4198:
are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
1887:) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a
347:). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance,
4683:
5539:
2989:
of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away
4537:
3503:
Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by
4316:
as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the
4696:
3905:{\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,}
4799:
5094:
1435:
1310:
3596:(and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the
1399:
radians). If the two complementary angles are adjacent, their non-shared sides form a right angle. In
Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a
4602:
2437:
subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the
1970:
The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the
5385:
5313:
285:, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples.
2235:
1924:
In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.
3560:
Metric
Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of
4303:
of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
1440:
4465:
2128:
207:
defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the
Neoplatonic metaphysician
6209:
5087:
5053:
4967:
4933:
3955:
4595:
5580:
of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as
6132:
331:
3987:
6565:
1986:
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be
5243:
5219:
5011:
4989:
2401:
being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the
1241:
4175:
In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an
2522:
turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 +
4007:
1705:
In
Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.
2794: turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called
5333:
2245:
thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio
4785:{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
3063:(albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2 equal parts for other values of
4248:
and the x-axis (positive or negative). Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude
997:
When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
2316:, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in
7714:
Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the
International System of Units (SI)".
4892:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
5184:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}
2964:
are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are
1586:{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\&\tan A=\cot B&&\sec A=\csc B\end{aligned}}}
2186:
5705:
has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The
2371: = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is
2085:
5594:
1702:
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
5248:
8164:
6716:
2141:
The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form
8052:
6595:
7335:
5663:, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.
6761:
2016:
in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.
1684:. However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a
8058:
4272:
correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).
4037:
Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal. For example, the
3557:
3288:
8159:
333:. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A").
2397:, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit
4678:{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
3574:
6682:
3617:
223:, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.
8132:
7705:
7667:
7504:
6979:
6559:
3961:
is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while
7408:
7057:
Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".
5534:{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}
288:
The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the
6686:
5588:
forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by
2630: = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 +
4901:
The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces
6172:
5058:
5024:
4938:
4904:
3915:
2755:
1046:
When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle
7922:
Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units".
6083:
This approach requires, however, an additional proof that the measure of the angle does not change with changing radius
1002:
A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called
7439:
2931:
of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128.
2313:
7162:
Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI".
2004:
of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full
211:, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by
144:, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
4568:
4532:{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}
4300:
140:. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a
17:
5645:
3124:
1875:. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an
4218:. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all
6646:
4308:
3467:
is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using
7513:
Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their
Pedagogic Value When Introduced Explicitly".
4098:
It is frequently helpful to impose a convention that allows positive and negative angular values to represent
8044:
6111:
4369:
at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—
7615:
4365:(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the
2538:= 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 +
2405:
display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the
299:
6900:
5612:
2443:
2442:
sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for
194:
7648:
7474:
4541:
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their
3972:
1714:
Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.
8039:
5973:
4110:
1892:
468:
An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an
53:
7912:
3379:
radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
3234:
subunits of the
Babylonian unit. It is straightforward to construct with ruler and compasses. It is the
1314:
I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.
5947:
5929:
3030:
1981:
8186:
6708:
6546:
4225:
Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called
4183:
passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
3059:. The binary degree is used in computing so that an angle can be efficiently represented in a single
2802:. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° =
2723:
5224:
5200:
2363:
is determined by the circumference of a circle that is equal in length to the radius of the circle (
502:
An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a
7281:
Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems".
4422:
4285:
holds. Some quantities related to angles where the angle addition postulate does not hold include:
4282:
3589:
2439:
1696:
232:
93:
that contains the rays. Angles are also formed by the intersection of two planes; these are called
7375:
5643:(that is, the apparent position of an astronomical object) can be identified using any of several
4994:
4972:
3496:. It is a long-established practice in mathematics and across all areas of science to make use of
7959:
Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem".
6463:
4321:
4038:
3116:
111:
6747:
6426:
6388:
2952:
The true milliradian is defined as a thousandth of a radian, which means that a rotation of one
8180:
7470:
6003:
5553:
4103:
4099:
3588:
At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a
3323:
2402:
2294:
1913:
In a triangle, three intersection points, each of an external angle bisector with the opposite
1197:
631:
260:
141:
38:
6969:
5741:
These measurements depend on the individual subject, and the above should be treated as rough
4065:
7494:
6013:
4214:
2876:
2270:
1884:
1814:
radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex
125:
73:
8062:
6580:
982:
Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.
8095:
8034:
7978:
7931:
7889:
7844:
7793:
7764:
7723:
7586:
7555:
7522:
7300:
7181:
7066:
7031:
6932:
6880:
6794:
6344:
5958:
5934:
5756:
are usually measured in angular units, expressed in terms of time, based on a 24-hour day.
4191:
3992:
3570:
3237:
3169:
2750:
is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2
2708:
2317:
2067:
1689:
1615:
983:
8:
6737:
6678:
5988:
5978:
5706:
5557:
5348:
4688:
4558:
1950:
1880:
1876:
8099:
7982:
7935:
7893:
7848:
7797:
7784:
Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle".
7768:
7727:
7590:
7559:
7526:
7304:
7185:
7070:
7035:
6936:
5649:, where the references vary according to the particular system. Astronomers measure the
3969:
which assumes its argument is a dimensionless number in radians. The capitalised symbol
8111:
7994:
7968:
7947:
7879:
7834:
7809:
7739:
7687:
7635:
7602:
7316:
7290:
7197:
7171:
6950:
6741:
6053:
5939:
5651:
5629:
5585:
5318:
4691:, the expression for the cosine above may give non-real values, so it is replaced with
4069:
3485:
2301:(i.e., the angle subtended by the circumference of a circle at its centre) is equal to
2274:
2135:
1776:
1414:, "to fill up". An acute angle is "filled up" by its complement to form a right angle.
397:
375:
There is some common terminology for angles, whose measure is always non-negative (see
60:
4421:
knew how to bisect an angle (divide it into two angles of equal measure) using only a
3468:
2956:
would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost all
2309:. Two exceptions are the radian (and its decimal submultiples) and the diameter part.
8128:
8115:
8107:
7998:
7951:
7943:
7813:
7805:
7752:
7743:
7701:
7663:
7639:
7631:
7606:
7598:
7500:
7435:
7320:
7201:
7193:
6975:
6954:
6555:
6518:
5666:
In both geography and astronomy, a sighting direction can be specified in terms of a
5569:
5565:
4418:
4313:
4180:
4113:, an angle is typically defined by its two sides, with its vertex at the origin. The
4059:
1964:
1673:
1025:
449:
220:
212:
120:
79:
7866:
Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022).
7857:
7822:
5953:
4408:
4125:
is defined by the measure from the initial side in radians, degrees, or turns, with
3414:
1305:{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} }
8127:, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163,
8123:
Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in
Intersecting and Parallel Lines",
8103:
8019:
7986:
7939:
7897:
7852:
7801:
7772:
7731:
7627:
7594:
7563:
7530:
7308:
7189:
7074:
7039:
7004:
6940:
6521:
6028:
5968:
5963:
5698:
5640:
5624:
of any location in terms of angles subtended at the center of the Earth, using the
5549:
5336:
4446:
4412:
4087:
3597:
3582:
3531:
3397:
was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a
2957:
1907:
292:
AB and AC (that is, the half-lines from point A through points B and C) is denoted
186:
90:
7616:"The next 50 years of the SI: a review of the opportunities for the e-Science age"
3626:. With this change the formula for the angle subtended at the center of a circle,
1200:
is a line that intersects a pair of (often parallel) lines and is associated with
7681:
7400:
6619:
6048:
6023:
5993:
5749:
5675:
5581:
5573:
4438:
3449:
is radius. One SI radian corresponds to the angle expressed in radians for which
3398:
3394:
3211:
3131:
2953:
2735:
2298:
1872:
1772:
1699:
from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
1677:
638:
137:
3530:
the unit radian does not appear in the result. Similarly in the formula for the
2008:
are effectively equivalent. In other contexts, such as identifying a point on a
1618:" in the names of some trigonometric ratios refers to the word "complementary".
8204:
7990:
7902:
7867:
7735:
7312:
5998:
5671:
5667:
5589:
4426:
2414:
2282:
2174:
1959:
1863:
1804:
1767:
1762:
1103:, either of these angle measures may be used to determine the measure of Angle
908:
289:
171:
129:
95:
68:
49:
6945:
6920:
4244:
in standard position is the positive acute angle between the terminal side of
1621:
562:
8198:
8155:
8150:
6033:
5983:
5742:
5728:
5352:
5194:
4542:
4176:
4154:
3647:
3593:
3566:
3562:
3484:
gives 1 SI radian as 1 m/m = 1. The key fact is that the SI radian is a
3047:
2715:
2662:
degrees, or 3.125 degrees. The arcsecond is the angle used to measure a
2555:
2166:
is the measure of a complete turn expressed in the chosen unit (for example,
1914:
1807:
1685:
461:
455:
216:
6886:
6860:
6800:
6778:
6541:
4208:
Angles that have the same measure (i.e., the same magnitude) are said to be
4074:
3610:
In particular, Quincey identifies
Torrens' proposal to introduce a constant
3172:. In German, the symbol has been used to denote a quadrant. 1 quad = 90° =
2000:
In some contexts, such as identifying a point on a circle or describing the
351:
always refers to the anticlockwise (positive) angle from B to C about A and
6995:
French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)".
6063:
6008:
5943:
4013:
3280:
2670:
2559:
2063:
2005:
1918:
991:
420:
236:
8024:
8007:
7677:
6043:
5765:
5753:
4457:
4083:
4046:
3966:
3522:
is the angle through which the pulley turns in radians. When multiplying
3231:
3227:
3164:
2936:
2712:
2699:. It is a decimal subunit of the quadrant. A right angle is 100 grads. A
2434:
554:
443:
364:
265:
2429:, denoted by a small superscript circle (°), is 1/360 of a turn, so one
1939:
supplement!) of the interior angle. This conflicts with the above usage.
1906:
of two exterior angles and the bisector of the other interior angle are
540:
The names, intervals, and measuring units are shown in the table below:
7577:
Eder, W E (January 1982). "A Viewpoint on the
Quantity "Plane Angle"".
7250:
6058:
6038:
6018:
5197:
can be extended to subspaces of finite dimensions. Given two subspaces
4546:
4187:
2896:
2763:
2451:
1954:
580:) angles. The acute and obtuse angles are also known as oblique angles.
363:"Oblique angle" redirects here. For the cinematographic technique, see
8179:
7534:
7078:
7043:
7008:
4429:
showed that this construction could not be performed for most angles.
8168:, vol. 2 (11th ed.), Cambridge University Press, p. 14
7356:
7238:
6526:
5702:
5636:
5621:
5607:
4158:
4091:
4049:'s unit system similarly considers angles to have an angle dimension.
2727:
2700:
2571:
2463:
2447:
1903:
1600:
8059:
University of Texas research department: linguistics research center
7776:
7568:
7543:
6657:
6359:
6357:
5709:
can convert such an angular measurement into a distance/size ratio.
5568:. The comparison can be visualized as the size of the openings of a
1692:(one whose vertices all fall on a single circle) are supplementary.
8086:
Torrens, A B (1 January 1986). "On Angles and Angular Quantities".
7973:
7884:
7295:
7176:
5617:
5610:, the location of any point on the Earth can be identified using a
5308:{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}
3616:
equal to 1 inverse radian (1 rad) in a fashion similar to the
2066:
centered at the vertex of the angle is drawn, e.g., with a pair of
1899:
1780:
1733:
The difference between an angle and a complete angle is termed the
1710:
1604:
1401:
1320:
269:
is typically not used for this purpose to avoid confusion with the
8149:
This article incorporates text from a publication now in the
7839:
3283:. It equals 6°, so a whole turn was divided into 60 hexacontades.
978:
8012:
Journal of Research of the National Bureau of Standards Section B
7022:
Oberhofer, E. S. (March 1992). "What happens to the 'radians'?".
6354:
5683:
5679:
5625:
5356:
4366:
4195:
2961:
2691:
2290:
2178:
1815:
1752:
1596:
1149:
937:
392:
An angle smaller than a right angle (less than 90°) is called an
208:
167:
104:
8077:
Shute, William G.; Shirk, William W.; Porter, George F. (1960),
7660:
Experiencing Geometry / Euclidean and Non-Euclidean with History
2019:
1417:
The difference between an angle and a right angle is termed the
7434:(6th ed.). Belmont, CA: Thomson Brooks/Cole. p. 110.
6542:
6472:
6340:
4337:
4249:
4130:
4118:
4079:
3578:
3120:
2663:
2343:
2323:
The following table lists some units used to represent angles.
2286:
2261:
2230:{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.}
2079:
2009:
1775:
has at least one interior angle, that is, a reflex angle. In
1695:
If a point P is exterior to a circle with center O, and if the
1611:
1029:
775:
586:
204:
133:
103:
may also define an angle, which is the angle of the rays lying
6863:, p. 151: "One radian corresponds to the angle for which
6781:, p. 151: "One radian corresponds to the angle for which
6654:
Manual & Technical Specifications - ooPIC Compiler Ver 6.0
5734:
10° is the approximate width of a closed fist at arm's length.
5021:
The definition of the angle between one-dimensional subspaces
4168:
is effectively equivalent to an angle of "one full turn minus
4106:
in opposite directions or "sense" relative to some reference.
2297:. Most units of angular measurement are defined such that one
548:
532:
An angle that is not a multiple of a right angle is called an
37:
This article is about angles in geometry. For other uses, see
6651:
6299:
6297:
5687:
5660:
4374:
4362:
4291:
4012:
Current SI can be considered relative to this framework as a
2704:
2058:
1404:
is 180 degrees, and the right angle accounts for 90 degrees.
190:
180:
174:
157:
100:
31:
2711:
of the Earth, so the kilometer is the decimal analog to the
5737:
20° is the approximate width of a handspan at arm's length.
5721:
5656:
5577:
4325:
4320:
Although done rarely, one can report the direct results of
3651:
3060:
2970:
1719:
Two angles that sum to a complete angle (1 turn, 360°, or 2
1317:
Three special angle pairs involve the summation of angles:
6294:
6139:
3612:
3424:
1867:; that is, an interior angle and an exterior angle form a
1771:
if it lies on the inside of that simple polygon. A simple
107:
to the respective curves at their point of intersection.
83:
of the angle. Angles formed by two rays are also known as
8054:
Preliminary Indo-European lexicon — Pokorny PIE data
6267:
5717:
1963:. It may be defined as the acute angle between two lines
1020:
The equality of vertically opposite angles is called the
990:"Vertical angle" redirects here. Not to be confused with
7865:
7686:, The Thirteen Books of Euclid's Elements, vol. 1,
7362:
7256:
7244:
7136:
7131:
6212:
1676:
and share just one side), their non-shared sides form a
355:
the anticlockwise (positive) angle from C to B about A.
44:
7056:
6516:
2406:
2131:
2123:{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .}
1365:
are angle pairs whose measures sum to one right angle (
1236:
states that if B is in the interior of angle AOC, then
270:
7821:
Mohr, Peter J; Phillips, William D (1 February 2015).
7496:
A High School First Course in Euclidean Plane Geometry
7228:
7226:
7213:
7211:
3492:. In SI 2019, the SI radian is defined accordingly as
1803:
turn; the measures of the interior angles of a simple
6369:
6284:
6282:
6175:
6114:
5388:
5321:
5251:
5227:
5203:
5097:
5061:
5027:
4997:
4975:
4941:
4907:
4802:
4699:
4605:
4571:
4468:
4141:-axis. When Cartesian coordinates are represented by
3995:
3975:
3918:
3664:
3489:
2293:(grad), though many others have been used throughout
2189:
2088:
1438:
1432:
are complementary, the following relationships hold:
1244:
302:
7662:(3rd ed.), Pearson Prentice Hall, p. 104,
6443:
5601:
5584:
in their angle argument, the circular ones are just
4391:(Gr. ξυστρίς, a tool for scraping), concavo-convex;
967:
77:
of the angle, sharing a common endpoint, called the
7262:
7223:
7208:
6204:{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}
5082:{\displaystyle \operatorname {span} (\mathbf {v} )}
5048:{\displaystyle \operatorname {span} (\mathbf {u} )}
4962:{\displaystyle \operatorname {span} (\mathbf {v} )}
4928:{\displaystyle \operatorname {span} (\mathbf {u} )}
3950:{\displaystyle x=\eta \theta =\theta /{\text{rad}}}
447:. Two lines that form a right angle are said to be
27:
Figure formed by two rays meeting at a common point
8190:, vol. 2 (9th ed.), 1878, pp. 29–30
6309:
6279:
6203:
6126:
5533:
5327:
5307:
5237:
5213:
5183:
5081:
5047:
5005:
4983:
4961:
4927:
4891:
4794:or, more commonly, using the absolute value, with
4784:
4677:
4589:
4531:
4432:
4402:
4001:
3981:
3957:is the angle in radians. The capitalized function
3949:
3904:
3230:, The degree, minute of arc and second of arc are
2229:
2122:
1585:
1304:
325:
7157:
7155:
7153:
7151:
6887:International Bureau of Weights and Measures 2019
6861:International Bureau of Weights and Measures 2019
6819:, the radian can be defined in terms of the area
6801:International Bureau of Weights and Measures 2019
6779:International Bureau of Weights and Measures 2019
5659:by imagining two lines through the center of the
4281:For an angular unit, it is definitional that the
2558:was historically defined as an arcminute along a
1861:The supplement of an interior angle is called an
8196:
8076:
7657:
7476:The Algebra of Coplanar Vectors and Trigonometry
7376:"UnityDimensions—Wolfram Language Documentation"
6656:. Savage Innovations, LLC. 2007 . Archived from
6478:
6410:
6408:
6363:
6104:Other proposals include the abbreviation "rad" (
5342:
4425:but could only trisect certain angles. In 1837,
4009:if it is clear that the complete form is meant.
6671:
6535:
3518:is the radius of the pulley in centimeters and
385:An angle equal to 0° or not turned is called a
7650:Elementary geometry: practical and theoretical
7646:
7148:
6971:Minds-on Physics: Advanced topics in mechanics
6754:
6303:
3363:(occasionally used in Islamic mathematics) is
2722: = 400). The grad is used mostly in
2134:, the radian is treated as being equal to the
1406:The adjective complementary is from the Latin
215:, who regarded an angle as a deviation from a
6647:"ooPIC Programmer's Guide - Chapter 15: URCP"
6572:
6511:CRC Standard Mathematical Tables and Formulae
6508:
6405:
4590:{\displaystyle \langle \cdot ,\cdot \rangle }
4345:is defined as the angle between the tangents
2281:, with the most contemporary units being the
1603:of its complement, and its secant equals the
7868:"On the dimension of angles and their units"
7820:
7753:"Dimensional angles and universal constants"
7750:
7658:Henderson, David W.; Taimina, Daina (2005),
7492:
7280:
7161:
7142:
7111:
6816:
6543:International Bureau of Weights and Measures
6273:
6147:
6121:
6115:
5595:Introduction to the Analysis of the Infinite
5119:
5103:
4824:
4808:
4727:
4711:
4622:
4606:
4584:
4572:
1688:are supplementary, and opposite angles of a
263:denoting the size of some angle (the symbol
7333:
5712:Other astronomical approximations include:
5016:
4137:representing rotations toward the negative
4129:representing rotations toward the positive
1779:, the measures of the interior angles of a
161:
7751:Lévy-Leblond, Jean-Marc (September 1998).
7653:(3rd ed.), Cambridge University Press
7541:
7107:
6907:Angular amplitude of swing No dimensions.
6905:. New Haven : Yale University Press.
6730:
6700:
6491:
6489:
6487:
6143:
1868:
1187:
8023:
7972:
7901:
7883:
7856:
7838:
7647:Godfrey, Charles; Siddons, A. W. (1919),
7567:
7512:
7294:
7175:
7091:
7021:
6944:
6760:
6683:"Angles, integers, and modulo arithmetic"
6105:
4561:, we replace the Euclidean dot product (
4041:units library defines angle units with a
2105:
1747:
1642:Two angles that sum to a straight angle (
336:In other ways, an angle denoted as, say,
128:conventionally defined as the ratio of a
8154:
8122:
8008:"Angle as a fourth fundamental quantity"
7454:
7429:
7334:Schabel, Matthias C.; Watanabe, Steven.
6898:
6641:
6639:
6578:
6513:, Boca Raton, FL: CRC Press, p. 270
6414:
6327:
6257:
5716:0.5° is the approximate diameter of the
5376:are the components of the metric tensor
5355:is used to define the angle between two
4336:
4332:
4073:
3577:(N⋅m/rad), and not in the quantities of
3558:American Association of Physics Teachers
2260:
2018:
1751:
1709:
1620:
1319:
1227:
1148:
1042:Equals subtracted from equals are equal.
977:
147:
43:
8085:
8081:, American Book Company, pp. 25–27
8032:
7958:
7921:
7713:
7675:
7458:
7268:
7232:
7217:
7127:
7123:
7103:
6967:
6851:), in which case it has the units m⋅m."
6815:, p. 844: "Also, as alluded to in
6812:
6484:
6261:
6235:
6166:
6127:{\displaystyle \langle \theta \rangle }
3407:
2130:Conventionally, in mathematics and the
1943:
1891:. Exterior angles are commonly used in
14:
8197:
8050:
8005:
7695:
7613:
7115:
7095:
6994:
6918:
6677:
6548:The International System of Units (SI)
6461:
6449:
6375:
6246:
6155:
6135:
3257: turn. 1 Babylonian unit = 60° =
326:{\displaystyle {\widehat {\rm {BAC}}}}
7910:
7783:
7119:
6736:
6706:
6636:
6517:
6315:
6288:
4557:To define angles in an abstract real
4276:
4179:, which is typically determined by a
4153:-axis upward, positive rotations are
2269:Throughout history, angles have been
986:are used here to show angle equality.
226:
7614:Foster, Marcus P (1 December 2010).
7576:
7411:from the original on 23 October 2017
7099:
6568:from the original on 18 October 2021
4379:, on both sides, κυρτός, convex) or
4341:The angle between the two curves at
4202:
3982:{\displaystyle \operatorname {Sin} }
3261:/3 rad ≈ 1.047197551 rad.
1668:If the two supplementary angles are
1107:. Using the measure of either angle
370:
185:) meaning "crooked, curved" and the
7700:, W. H. Freeman, pp. 97, 255,
5727:1° is the approximate width of the
5543:
2433:is 360°. One advantage of this old
1975:
1838: − 2)2 right angles, or (
509:An angle equal to 1 turn (360° or 2
219:; the second, angle as quality, by
24:
7544:"Angles—Let's treat them squarely"
6581:"On Angles and Angle Measurements"
6339:
6196:
6178:
5693:Astronomers also measure objects'
5291:
5263:
5230:
5206:
4023:is assumed to hold, or similarly,
2314:International System of Quantities
2113:
2110:
2107:
1834: − 2)180 degrees, (
1298:
1295:
1292:
1288:
1278:
1275:
1272:
1268:
1258:
1255:
1252:
1248:
1065:. Similarly, the measure of angle
377:
343:
313:
310:
307:
25:
8216:
8172:
7493:Aboughantous, Charles H. (2010),
6899:Bridgman, Percy Williams (1922).
6462:Willis, Clarence Addison (1922).
6321:
5602:Angles in geography and astronomy
4460:and their lengths by the formula
3548:, radians appear in the units of
2379: radians, and one radian is
1953:(such as two adjacent faces of a
1039:Equals added to equals are equal.
259:, . . . ) as
8142:
8006:Romain, Jacques E. (July 1962).
5315:, this leads to a definition of
5173:
5160:
5115:
5107:
5072:
5038:
4999:
4977:
4952:
4918:
4878:
4865:
4820:
4812:
4771:
4758:
4723:
4715:
4664:
4651:
4618:
4610:
4552:
4518:
4505:
4478:
4470:
4312:between two lines is defined in
4066:Sign (mathematics) § Angles
4053:
3413:This section is an excerpt from
3057:binary angular measurement (BAM)
2138:1, thus being normally omitted.
2012:curve or describing an object's
1931:of a simple polygon to mean the
1095:is supplementary to both angles
585:
561:
547:
48:A green angle formed by two red
7823:"Dimensionless units in the SI"
7542:Brownstein, K. R. (July 1997).
7486:
7464:
7448:
7423:
7393:
7368:
7327:
7274:
7085:
7050:
7015:
6988:
6961:
6919:Prando, Giacomo (August 2020).
6912:
6892:
6854:
6806:
6772:
6719:from the original on 2019-08-06
6689:from the original on 2019-06-30
6612:
6601:from the original on 2019-01-17
6502:
6455:
6419:
6381:
6098:
6077:
5646:astronomical coordinate systems
4433:Dot product and generalisations
4409:Bisection § Angle bisector
4403:Bisecting and trisecting angles
4361:The angle between a line and a
4164:In many contexts, an angle of −
4090:direction, and negative in the
3965:is the traditional function on
3441:is subtended angle in radians,
3162: turn and also known as a
3125:IEEE 754 recommended operations
2078:of the circle is the number of
1115:, we find the measure of angle
1091:and are congruent. Since angle
271:constant denoted by that symbol
193:". Both are connected with the
7676:Heiberg, Johan Ludvig (1908),
6743:The Growth of Physical Science
6579:Dimitrić, Radoslav M. (2012).
6364:Shute, Shirk & Porter 1960
6333:
6251:
6240:
6229:
5296:
5286:
5268:
5258:
5238:{\displaystyle {\mathcal {W}}}
5214:{\displaystyle {\mathcal {U}}}
5177:
5169:
5164:
5156:
5147:
5141:
5076:
5068:
5042:
5034:
4956:
4948:
4922:
4914:
4882:
4874:
4869:
4861:
4852:
4846:
4775:
4767:
4762:
4754:
4750:
4744:
4668:
4660:
4655:
4647:
4640:
4634:
4522:
4514:
4509:
4501:
4497:
4491:
3873:
3863:
3837:
3827:
3801:
3791:
3553:"pedagogically unsatisfying".
3421:Plane angle may be defined as
3168:. The quadrant is the unit in
2703:was historically defined as a
1895:when drawing regular polygons.
1036:All straight angles are equal.
201:, meaning "to bend" or "bow".
13:
1:
7430:McKeague, Charles P. (2008).
6974:. Kendall Hunt. p. 262.
6468:. Blakiston's Son. p. 8.
6222:
5343:Angles in Riemannian geometry
4549:from their vector equations.
4383:(Gr. κισσός, ivy), biconvex;
3618:introduction of the constant
3469:the area of a circular sector
3415:Radian § Dimensional analysis
3115:) unit is implemented in the
2305:units, for some whole number
1669:
273:). Lower case Roman letters (
7690:: Cambridge University Press
7401:"Mathwords: Reference Angle"
6968:Leonard, William J. (1999).
6707:Bonin, Walter (2016-01-11).
6479:Henderson & Taimina 2005
5616:. This system specifies the
5613:geographic coordinate system
5006:{\displaystyle \mathbf {v} }
4984:{\displaystyle \mathbf {u} }
4419:ancient Greek mathematicians
4070:Euclidean space § Angle
1889:supplementary exterior angle
1756:Internal and external angles
1188:§ Combining angle pairs
181:
67:is the figure formed by two
7:
8040:Encyclopedia of Mathematics
7757:American Journal of Physics
7548:American Journal of Physics
7515:American Journal of Physics
7351:Angles are treated as units
6588:The Teaching of Mathematics
6509:D. Zwillinger, ed. (1995),
6499:, Dover Publications, 2007.
6497:Advanced Euclidean Geometry
5974:Argument (complex analysis)
5922:
5335:angles called canonical or
4111:Cartesian coordinate system
2265:Definition of 1 radian
1761:An angle that is part of a
1680:. Such angles are called a
1410:, associated with the verb
1222:consecutive interior angles
472:("obtuse" meaning "blunt").
54:Cartesian coordinate system
10:
8221:
8108:10.1088/0026-1394/22/1/002
7944:10.1088/0026-1394/53/2/840
7914:Modern Elementary Geometry
7806:10.1088/0026-1394/53/3/991
7696:Jacobs, Harold R. (1974),
7632:10.1088/0026-1394/47/6/R01
7599:10.1088/0026-1394/18/1/002
7336:"Boost.Units FAQ – 1.79.0"
7194:10.1088/0026-1394/53/3/998
6762:Murnaghan, Francis Dominic
6304:Godfrey & Siddons 1919
5930:Angle measuring instrument
4406:
4375:
4157:, and negative cycles are
4121:, while the other side or
4063:
4057:
4035:in mathematical formulas.
4016:system where the equation
3412:
3326:unit equal to about 2° or
2915:of a turn. 1 point =
2070:. The ratio of the length
1982:Angle measuring instrument
1979:
1927:Some authors use the name
1146:and are equal in measure.
1014:. They are abbreviated as
1012:vertically opposite angles
989:
630:
362:
175:
36:
29:
8051:Slocum, Jonathan (2007),
7858:10.1088/0026-1394/52/1/40
6946:10.1038/s41567-020-0997-3
4395:(Gr. κοίλη, a hollow) or
4086:count as positive in the
3226:was the unit used by the
2074:of the arc by the radius
1910:(meet at a single point).
1214:alternate interior angles
1210:alternate exterior angles
114:of an angle is called an
8079:Plane and Solid Geometry
8033:Sidorov, L. A. (2001) ,
7991:10.1088/1681-7575/ac023f
7911:Moser, James M. (1971),
7903:10.1088/1681-7575/ac7bc2
7736:10.1088/1681-7575/abe0fc
7499:, Universal Publishers,
7313:10.1088/1681-7575/aa7160
7143:Mohr & Phillips 2015
6817:Mohr & Phillips 2015
6070:
5367:are tangent vectors and
5017:Angles between subspaces
4423:compass and straightedge
4283:angle addition postulate
4149:-axis rightward and the
3636:, is modified to become
3590:base unit of measurement
3461:1 SI radian = 1 m/m
2444:geographical coordinates
2256:
1933:explement exterior angle
1234:angle addition postulate
1134:. Therefore, both angle
1028:attributed the proof to
358:
233:mathematical expressions
30:Not to be confused with
8187:Encyclopædia Britannica
8165:Encyclopædia Britannica
8125:New Century Mathematics
6746:. CUP Archive. p.
5639:, a given point on the
4969:spanned by the vectors
4565:) by the inner product
4322:trigonometric functions
4031:allows the omission of
2610:of a minute of arc and
2403:trigonometric functions
1599:of an angle equals the
1169:, often abbreviated as
1142:have measures equal to
1084:have measures equal to
1054:, the measure of angle
7471:Robert Baldwin Hayward
6427:"Supplementary Angles"
6389:"Complementary Angles"
6205:
6128:
6004:Exterior angle theorem
5535:
5329:
5309:
5239:
5215:
5185:
5083:
5049:
5007:
4985:
4963:
4929:
4893:
4786:
4679:
4591:
4533:
4358:
4222:are equal in measure).
4095:
4003:
3983:
3951:
3906:
3206:turn = 100 grad.
3119:scientific calculator
2566: = 21,600).
2266:
2231:
2124:
2053:
1949:The angle between two
1757:
1748:Polygon-related angles
1715:
1638:
1587:
1357:
1306:
1162:
1022:vertical angle theorem
987:
327:
235:, it is common to use
162:
56:
39:Angle (disambiguation)
8025:10.6028/jres.066B.012
7380:reference.wolfram.com
6709:"RE: WP-32S in 2016?"
6624:TheFreeDictionary.com
6206:
6138:), and the constants
6129:
6014:Great circle distance
5745:approximations only.
5724:as viewed from Earth.
5564:is the argument of a
5536:
5330:
5310:
5240:
5216:
5186:
5084:
5050:
5008:
4986:
4964:
4930:
4894:
4787:
4680:
4592:
4534:
4340:
4333:Angles between curves
4109:In a two-dimensional
4077:
4004:
4002:{\displaystyle \sin }
3984:
3952:
3907:
2707:-grad of arc along a
2440:"minute" and "second"
2277:. These are known as
2264:
2232:
2125:
2023:The measure of angle
2022:
1869:linear pair of angles
1755:
1713:
1682:linear pair of angles
1672:(i.e., have a common
1624:
1588:
1352:is the complement of
1323:
1307:
1228:Combining angle pairs
1152:
981:
513:radians) is called a
495:radians) is called a
441:radians) is called a
328:
148:History and etymology
47:
6902:Dimensional analysis
6738:Jeans, James Hopwood
6328:Wong & Wong 2009
6173:
6112:
5959:Angular acceleration
5935:Angles between flats
5678:with respect to the
5386:
5319:
5249:
5225:
5201:
5095:
5059:
5025:
4995:
4973:
4939:
4905:
4800:
4697:
4603:
4569:
4466:
4456:is related to their
3993:
3973:
3916:
3662:
3571:angular acceleration
3534:of a rolling wheel,
3408:Dimensional analysis
3238:equilateral triangle
3045:, also known as the
2318:dimensional analysis
2187:
2086:
2056:To measure an angle
1944:Plane-related angles
1893:Logo Turtle programs
1723:radians) are called
1690:cyclic quadrilateral
1664:supplementary angles
1662:radians) are called
1607:of its complement.)
1436:
1363:Complementary angles
1242:
1218:corresponding angles
491: turn (180° or
378:§ Signed angles
344:§ Signed angles
300:
166:, meaning "corner".
8100:1986Metro..22....1T
7983:2021Metro..58e3002Q
7936:2016Metro..53..840Q
7894:2022Metro..59e3001M
7849:2015Metro..52...40M
7798:2016Metro..53..991M
7769:1998AmJPh..66..814L
7728:2021Metro..58e2001L
7591:1982Metro..18....1E
7560:1997AmJPh..65..605B
7527:1936AmJPh...4..175B
7305:2017Metro..54..454Q
7186:2016Metro..53..998Q
7071:1993PhTea..31...84A
7059:The Physics Teacher
7036:1992PhTea..30..170O
7024:The Physics Teacher
6997:The Physics Teacher
6937:2020NatPh..16..888P
5989:Clock angle problem
5979:Astrological aspect
5707:small-angle formula
5701:. For example, the
5558:hyperbolic function
5349:Riemannian geometry
5339:between subspaces.
4689:inner product space
4559:inner product space
4117:is on the positive
4078:Measuring from the
3575:torsional stiffness
3514:centimeters, where
3445:is arc length, and
2973:mil" is defined as
2014:cumulative rotation
1830: radians, or (
1725:explementary angles
195:Proto-Indo-European
118:or simply "angle".
99:. Two intersecting
89:as they lie in the
6685:. blogs.msdn.com.
6519:Weisstein, Eric W.
6495:Johnson, Roger A.
6431:www.mathsisfun.com
6393:www.mathsisfun.com
6330:, pp. 161–163
6264:, pp. 177–178
6201:
6124:
6054:Transcendent angle
5948:standard deviation
5940:Angular statistics
5652:angular separation
5630:Greenwich meridian
5628:and (usually) the
5586:alternating series
5531:
5325:
5305:
5235:
5211:
5181:
5079:
5045:
5003:
4981:
4959:
4925:
4889:
4782:
4675:
4587:
4529:
4359:
4277:Related quantities
4236:(sometimes called
4096:
3999:
3979:
3947:
3902:
3486:dimensionless unit
3279:is a unit used by
2458: = 360)
2332:Number in one turn
2267:
2227:
2136:dimensionless unit
2120:
2054:
1787:radians, 180°, or
1777:Euclidean geometry
1758:
1716:
1639:
1583:
1581:
1358:
1302:
1163:
1125:) = 180° − 180° +
988:
475:An angle equal to
403:An angle equal to
396:("acute" meaning "
323:
227:Identifying angles
170:words include the
61:Euclidean geometry
57:
8134:978-0-19-800177-5
7707:978-0-7167-0456-0
7669:978-0-13-143748-7
7535:10.1119/1.1999110
7506:978-1-59942-822-2
7405:www.mathwords.com
7112:Lévy-Leblond 1998
7079:10.1119/1.2343667
7044:10.1119/1.2343500
7009:10.1119/1.2343535
6981:978-0-7872-5412-4
6921:"A spectral unit"
6766:Analytic Geometry
6679:Hargreaves, Shawn
6561:978-92-822-2272-0
6366:, pp. 25–27.
6351:Proposition I:13.
6274:Aboughantous 2010
6199:
6148:Lévy-Leblond 1998
5920:
5919:
5570:hyperbolic sector
5566:circular function
5526:
5525:
5328:{\displaystyle k}
5013:correspondingly.
4645:
4447:Euclidean vectors
4397:angulus lunularis
4314:rational geometry
4227:coterminal angles
4203:Equivalent angles
4145:, defined by the
4143:standard position
4060:Angle of rotation
4029:radian convention
3945:
3891:
3855:
3819:
3768:
3743:
3718:
3685:
3405:
3404:
3393:In old Arabia, a
3170:Euclid's Elements
3009:= 0.0009817... ≈
2774:The astronomical
2222:
2209:
2103:
1822:sides add up to (
1030:Thales of Miletus
1026:Eudemus of Rhodes
965:
964:
929:(180, 360)°
602:Name
576:), and straight (
371:Individual angles
320:
221:Carpus of Antioch
213:Eudemus of Rhodes
121:Angle of rotation
16:(Redirected from
8212:
8191:
8183:
8169:
8148:
8146:
8145:
8137:
8119:
8082:
8073:
8072:
8070:
8061:, archived from
8047:
8029:
8027:
8002:
7976:
7955:
7918:
7907:
7905:
7887:
7862:
7860:
7842:
7817:
7780:
7747:
7710:
7691:
7672:
7654:
7643:
7610:
7573:
7571:
7538:
7509:
7480:
7468:
7462:
7452:
7446:
7445:
7427:
7421:
7420:
7418:
7416:
7397:
7391:
7390:
7388:
7386:
7372:
7366:
7363:Mohr et al. 2022
7360:
7354:
7353:
7348:
7346:
7331:
7325:
7324:
7298:
7278:
7272:
7266:
7260:
7257:Mohr et al. 2022
7254:
7248:
7245:Mohr et al. 2022
7242:
7236:
7230:
7221:
7215:
7206:
7205:
7179:
7159:
7146:
7140:
7134:
7132:Mohr et al. 2022
7089:
7083:
7082:
7054:
7048:
7047:
7019:
7013:
7012:
6992:
6986:
6985:
6965:
6959:
6958:
6948:
6916:
6910:
6909:
6896:
6890:
6884:
6878:
6876:
6872:
6858:
6852:
6850:
6843:
6841:
6840:
6837:
6834:
6810:
6804:
6798:
6792:
6790:
6776:
6770:
6769:
6758:
6752:
6751:
6734:
6728:
6727:
6725:
6724:
6704:
6698:
6697:
6695:
6694:
6675:
6669:
6668:
6666:
6665:
6643:
6634:
6633:
6631:
6630:
6616:
6610:
6609:
6607:
6606:
6600:
6585:
6576:
6570:
6569:
6554:(9th ed.),
6553:
6539:
6533:
6532:
6531:
6522:"Exterior Angle"
6514:
6506:
6500:
6493:
6482:
6476:
6470:
6469:
6459:
6453:
6447:
6441:
6440:
6438:
6437:
6423:
6417:
6412:
6403:
6402:
6400:
6399:
6385:
6379:
6373:
6367:
6361:
6352:
6350:
6337:
6331:
6325:
6319:
6313:
6307:
6301:
6292:
6286:
6277:
6271:
6265:
6255:
6249:
6244:
6238:
6233:
6216:
6213:Mohr et al. 2022
6210:
6208:
6207:
6202:
6200:
6195:
6187:
6182:
6181:
6133:
6131:
6130:
6125:
6108:), the notation
6102:
6090:
6088:
6081:
6029:Irrational angle
5969:Angular velocity
5964:Angular diameter
5915:
5914:
5910:
5903:
5902:
5898:
5891:
5890:
5886:
5885:
5866:
5865:
5861:
5854:
5853:
5849:
5842:
5841:
5837:
5836:
5815:
5814:
5810:
5803:
5802:
5798:
5797:
5759:
5758:
5731:at arm's length.
5699:angular diameter
5686:with respect to
5641:celestial sphere
5550:hyperbolic angle
5544:Hyperbolic angle
5540:
5538:
5537:
5532:
5527:
5524:
5520:
5519:
5518:
5509:
5508:
5499:
5498:
5481:
5477:
5476:
5475:
5466:
5465:
5456:
5455:
5438:
5437:
5436:
5435:
5426:
5425:
5416:
5415:
5402:
5337:principal angles
5334:
5332:
5331:
5326:
5314:
5312:
5311:
5306:
5295:
5294:
5267:
5266:
5244:
5242:
5241:
5236:
5234:
5233:
5220:
5218:
5217:
5212:
5210:
5209:
5190:
5188:
5187:
5182:
5180:
5176:
5167:
5163:
5154:
5150:
5126:
5122:
5118:
5110:
5088:
5086:
5085:
5080:
5075:
5054:
5052:
5051:
5046:
5041:
5012:
5010:
5009:
5004:
5002:
4990:
4988:
4987:
4982:
4980:
4968:
4966:
4965:
4960:
4955:
4934:
4932:
4931:
4926:
4921:
4898:
4896:
4895:
4890:
4885:
4881:
4872:
4868:
4859:
4855:
4831:
4827:
4823:
4815:
4791:
4789:
4788:
4783:
4778:
4774:
4765:
4761:
4734:
4730:
4726:
4718:
4684:
4682:
4681:
4676:
4671:
4667:
4658:
4654:
4643:
4621:
4613:
4596:
4594:
4593:
4588:
4538:
4536:
4535:
4530:
4525:
4521:
4512:
4508:
4481:
4473:
4413:Angle trisection
4378:
4377:
4299:is equal to the
4270:
4266:
4264:
4263:
4260:
4257:
4240:) for any angle
4088:counterclockwise
4082:, angles on the
4044:
4034:
4026:
4022:
4008:
4006:
4005:
4000:
3988:
3986:
3985:
3980:
3964:
3960:
3956:
3954:
3953:
3948:
3946:
3943:
3941:
3911:
3909:
3908:
3903:
3892:
3890:
3882:
3881:
3880:
3861:
3856:
3854:
3846:
3845:
3844:
3825:
3820:
3818:
3810:
3809:
3808:
3789:
3769:
3767:
3759:
3758:
3749:
3744:
3742:
3734:
3733:
3724:
3719:
3717:
3709:
3708:
3699:
3683:
3657:
3645:
3635:
3615:
3606:
3598:area of a circle
3583:angular momentum
3551:
3547:
3532:angular velocity
3529:
3525:
3521:
3517:
3513:
3499:
3495:
3483:
3466:
3462:
3458:
3448:
3444:
3440:
3436:
3378:
3376:
3375:
3372:
3369:
3344:
3342:
3341:
3338:
3335:
3331:
3314:
3312:
3311:
3308:
3305:
3301:
3260:
3256:
3254:
3253:
3250:
3247:
3205:
3203:
3202:
3199:
3196:
3189:
3187:
3186:
3183:
3180:
3179:
3161:
3159:
3158:
3155:
3152:
3114:
3108:
3093:
3084:
3082:
3081:
3078:
3075:
3024:
3022:
3021:
3018:
3015:
3008:
3006:
3005:
3002:
2999:
2998:
2988:
2986:
2985:
2982:
2979:
2946:
2930:
2928:
2927:
2924:
2921:
2914:
2912:
2911:
2908:
2905:
2870:
2868:
2867:
2864:
2861:
2857:
2851:
2849:
2848:
2845:
2842:
2835:
2833:
2832:
2829:
2826:
2819:
2817:
2816:
2813:
2810:
2809:
2793:
2791:
2790:
2787:
2784:
2753:
2726:and continental
2661:
2659:
2658:
2655:
2652:
2645:
2643:
2642:
2639:
2636:
2625:
2623:
2622:
2619:
2616:
2609:
2607:
2606:
2603:
2600:
2553:
2551:
2550:
2547:
2544:
2537:
2535:
2534:
2531:
2528:
2521:
2519:
2518:
2515:
2512:
2505:
2503:
2502:
2499:
2496:
2396:
2394:
2393:
2392:
2388:
2385:
2378:
2370:
2353:
2326:
2325:
2244:
2236:
2234:
2233:
2228:
2223:
2215:
2210:
2208:
2197:
2177:or 400 grad for
2172:
2161:
2159:
2158:
2157:
2152:
2149:
2129:
2127:
2126:
2121:
2116:
2104:
2096:
2051:
2049:
2047:
2046:
2041:
2038:
2028:
1995:equal in measure
1976:Measuring angles
1857:
1855:
1854:
1851:
1848:
1829:
1813:
1802:
1800:
1799:
1796:
1793:
1786:
1737:of the angle or
1729:conjugate angles
1722:
1661:
1657:
1655:
1654:
1651:
1648:
1592:
1590:
1589:
1584:
1582:
1556:
1532:
1516:
1515:
1497:
1496:
1486:
1472:
1471:
1453:
1452:
1442:
1398:
1396:
1395:
1392:
1389:
1388:
1380:
1378:
1377:
1374:
1371:
1311:
1309:
1308:
1303:
1301:
1281:
1261:
1133:
1090:
1075:
1064:
973:
972:
958:(200, 400)
952:(100, 200)
923:(90, 180)°
903:
900:
892:
889:
883:
875:
872:
865:
862:
856:
852:
850:
849:
846:
843:
833:
830:
826:
824:
823:
820:
817:
808:
805:
801:
799:
798:
795:
792:
782:
770:
765:
763:
761:
760:
757:
754:
744:
742:
740:
739:
736:
733:
724:
722:
720:
719:
716:
713:
706:
704:
703:
700:
697:
687:
685:
683:
682:
679:
676:
667:
665:
663:
662:
659:
656:
646:
599:
598:
589:
565:
551:
512:
494:
490:
488:
487:
484:
481:
440:
438:
437:
434:
431:
430:
418:
416:
415:
412:
409:
354:
350:
339:
332:
330:
329:
324:
322:
321:
316:
305:
295:
268:
184:
178:
177:
165:
21:
18:Angle (geometry)
8220:
8219:
8215:
8214:
8213:
8211:
8210:
8209:
8195:
8194:
8178:
8175:
8158:, ed. (1911), "
8143:
8141:
8135:
8068:
8066:
8065:on 27 June 2010
7917:, Prentice-Hall
7777:10.1119/1.18964
7708:
7670:
7569:10.1119/1.18616
7507:
7489:
7484:
7483:
7469:
7465:
7453:
7449:
7442:
7428:
7424:
7414:
7412:
7399:
7398:
7394:
7384:
7382:
7374:
7373:
7369:
7361:
7357:
7344:
7342:
7332:
7328:
7279:
7275:
7267:
7263:
7259:, pp. 8–9.
7255:
7251:
7243:
7239:
7231:
7224:
7216:
7209:
7170:(3): 998–1002.
7160:
7149:
7141:
7137:
7108:Brownstein 1997
7090:
7086:
7055:
7051:
7020:
7016:
6993:
6989:
6982:
6966:
6962:
6917:
6913:
6897:
6893:
6885:
6881:
6874:
6864:
6859:
6855:
6838:
6835:
6832:
6831:
6829:
6824:
6811:
6807:
6799:
6795:
6782:
6777:
6773:
6759:
6755:
6735:
6731:
6722:
6720:
6705:
6701:
6692:
6690:
6676:
6672:
6663:
6661:
6645:
6644:
6637:
6628:
6626:
6618:
6617:
6613:
6604:
6602:
6598:
6583:
6577:
6573:
6562:
6551:
6545:(20 May 2019),
6540:
6536:
6507:
6503:
6494:
6485:
6477:
6473:
6460:
6456:
6448:
6444:
6435:
6433:
6425:
6424:
6420:
6413:
6406:
6397:
6395:
6387:
6386:
6382:
6374:
6370:
6362:
6355:
6338:
6334:
6326:
6322:
6314:
6310:
6302:
6295:
6287:
6280:
6272:
6268:
6256:
6252:
6245:
6241:
6234:
6230:
6225:
6220:
6219:
6188:
6186:
6177:
6176:
6174:
6171:
6170:
6164:
6144:Brownstein 1997
6113:
6110:
6109:
6103:
6099:
6094:
6093:
6084:
6082:
6078:
6073:
6068:
6049:Spherical angle
6024:Inscribed angle
5994:Decimal degrees
5925:
5912:
5908:
5907:
5900:
5896:
5895:
5888:
5883:
5882:
5881:
5863:
5859:
5858:
5851:
5847:
5846:
5839:
5834:
5833:
5832:
5812:
5808:
5807:
5800:
5795:
5794:
5793:
5750:right ascension
5682:as well as the
5632:as references.
5604:
5582:infinite series
5574:circular sector
5546:
5514:
5510:
5504:
5500:
5491:
5487:
5486:
5482:
5471:
5467:
5461:
5457:
5448:
5444:
5443:
5439:
5431:
5427:
5421:
5417:
5408:
5404:
5403:
5401:
5387:
5384:
5383:
5375:
5345:
5320:
5317:
5316:
5290:
5289:
5262:
5261:
5250:
5247:
5246:
5229:
5228:
5226:
5223:
5222:
5205:
5204:
5202:
5199:
5198:
5172:
5168:
5159:
5155:
5134:
5130:
5114:
5106:
5102:
5098:
5096:
5093:
5092:
5071:
5060:
5057:
5056:
5037:
5026:
5023:
5022:
5019:
4998:
4996:
4993:
4992:
4976:
4974:
4971:
4970:
4951:
4940:
4937:
4936:
4917:
4906:
4903:
4902:
4877:
4873:
4864:
4860:
4839:
4835:
4819:
4811:
4807:
4803:
4801:
4798:
4797:
4770:
4766:
4757:
4753:
4722:
4714:
4710:
4706:
4698:
4695:
4694:
4663:
4659:
4650:
4646:
4617:
4609:
4604:
4601:
4600:
4570:
4567:
4566:
4555:
4517:
4513:
4504:
4500:
4477:
4469:
4467:
4464:
4463:
4439:Euclidean space
4435:
4415:
4407:Main articles:
4405:
4356:
4352:
4348:
4335:
4279:
4268:
4267:turn, 180°, or
4261:
4258:
4255:
4254:
4252:
4234:reference angle
4205:
4135:negative angles
4127:positive angles
4072:
4062:
4056:
4051:
4050:
4045:dimension, and
4042:
4032:
4024:
4017:
3994:
3991:
3990:
3989:can be denoted
3974:
3971:
3970:
3962:
3958:
3942:
3937:
3917:
3914:
3913:
3883:
3876:
3872:
3862:
3860:
3847:
3840:
3836:
3826:
3824:
3811:
3804:
3800:
3790:
3788:
3760:
3754:
3750:
3748:
3735:
3729:
3725:
3723:
3710:
3704:
3700:
3698:
3663:
3660:
3659:
3655:
3637:
3627:
3624:
3611:
3601:
3549:
3535:
3527:
3523:
3519:
3515:
3505:
3497:
3493:
3471:
3464:
3460:
3450:
3446:
3442:
3438:
3422:
3418:
3410:
3373:
3370:
3367:
3366:
3364:
3339:
3336:
3333:
3332:
3329:
3327:
3309:
3306:
3303:
3302:
3299:
3297:
3258:
3251:
3248:
3245:
3244:
3242:
3200:
3197:
3194:
3193:
3191:
3184:
3181:
3177:
3176:
3175:
3173:
3156:
3153:
3150:
3149:
3147:
3112:
3106:
3091:
3079:
3076:
3073:
3072:
3070:
3019:
3016:
3013:
3012:
3010:
3003:
3000:
2996:
2994:
2993:
2991:
2983:
2980:
2977:
2976:
2974:
2941:
2925:
2922:
2919:
2918:
2916:
2909:
2906:
2903:
2902:
2900:
2877:(compass) point
2865:
2862:
2859:
2858:
2855:
2853:
2846:
2843:
2840:
2839:
2837:
2830:
2827:
2824:
2823:
2821:
2814:
2811:
2807:
2806:
2805:
2803:
2788:
2785:
2782:
2781:
2779:
2751:
2656:
2653:
2650:
2649:
2647:
2640:
2637:
2634:
2633:
2631:
2620:
2617:
2614:
2613:
2611:
2604:
2601:
2598:
2597:
2595:
2562:of the Earth. (
2548:
2545:
2542:
2541:
2539:
2532:
2529:
2526:
2525:
2523:
2516:
2513:
2510:
2509:
2507:
2500:
2497:
2494:
2493:
2491:
2390:
2389:
2386:
2383:
2382:
2380:
2376:
2375:. One turn is 2
2368:
2348:
2289:(rad), and the
2259:
2240:
2214:
2201:
2196:
2188:
2185:
2184:
2167:
2155:
2153:
2150:
2145:
2144:
2142:
2106:
2095:
2087:
2084:
2083:
2077:
2073:
2061:
2042:
2039:
2034:
2033:
2031:
2030:
2024:
1984:
1978:
1946:
1852:
1849:
1846:
1845:
1843:
1842: − 2)
1827:
1826: − 2)
1811:
1797:
1794:
1791:
1790:
1788:
1784:
1773:concave polygon
1750:
1745:
1720:
1659:
1658:turn, 180°, or
1652:
1649:
1646:
1645:
1643:
1632:
1628:
1580:
1579:
1555:
1530:
1529:
1511:
1507:
1492:
1488:
1485:
1467:
1463:
1448:
1444:
1439:
1437:
1434:
1433:
1393:
1390:
1386:
1385:
1384:
1382:
1375:
1372:
1369:
1368:
1366:
1355:
1351:
1347:
1339:
1335:
1331:
1291:
1271:
1251:
1243:
1240:
1239:
1230:
1206:interior angles
1202:exterior angles
1194:
1167:Adjacent angles
1121:180° − (180° −
1120:
1085:
1070:
1059:
1008:opposite angles
1004:vertical angles
995:
976:
970:
969:
898:
895:
887:
881:
878:
870:
868:
860:
854:
847:
844:
841:
840:
838:
836:
828:
821:
818:
815:
814:
812:
811:
803:
796:
793:
790:
789:
787:
785:
780:
768:
758:
755:
752:
751:
749:
747:
737:
734:
731:
730:
728:
727:
717:
714:
711:
710:
708:
701:
698:
695:
694:
692:
690:
680:
677:
674:
673:
671:
670:
660:
657:
654:
653:
651:
649:
644:
617:straight angle
597:
596:
595:
594:
593:
590:
582:
581:
579:
575:
571:
566:
558:
557:
552:
510:
492:
485:
482:
479:
478:
476:
435:
432:
428:
427:
426:
424:
413:
410:
407:
406:
404:
373:
368:
361:
352:
348:
337:
306:
304:
303:
301:
298:
297:
293:
264:
258:
254:
250:
246:
242:
229:
156:comes from the
150:
138:negative number
136:, and may be a
116:angular measure
96:dihedral angles
42:
35:
28:
23:
22:
15:
12:
11:
5:
8218:
8208:
8207:
8193:
8192:
8174:
8173:External links
8171:
8156:Chisholm, Hugh
8139:
8138:
8133:
8120:
8083:
8074:
8048:
8030:
8003:
7956:
7930:(2): 840–845.
7919:
7908:
7863:
7818:
7792:(3): 991–997.
7781:
7763:(9): 814–815.
7748:
7711:
7706:
7693:
7673:
7668:
7655:
7644:
7626:(6): R41–R51.
7611:
7574:
7554:(7): 605–614.
7539:
7521:(4): 175–179.
7510:
7505:
7488:
7485:
7482:
7481:
7463:
7447:
7441:978-0495382607
7440:
7422:
7392:
7367:
7355:
7326:
7289:(4): 454–460.
7273:
7261:
7249:
7237:
7222:
7207:
7147:
7135:
7092:Brinsmade 1936
7084:
7049:
7030:(3): 170–171.
7014:
7003:(5): 260–261.
6987:
6980:
6960:
6925:Nature Physics
6911:
6891:
6889:, p. 137.
6879:
6853:
6805:
6803:, p. 151.
6793:
6771:
6753:
6729:
6699:
6670:
6635:
6620:"angular unit"
6611:
6594:(2): 133–140.
6571:
6560:
6534:
6501:
6483:
6481:, p. 104.
6471:
6465:Plane Geometry
6454:
6442:
6418:
6404:
6380:
6378:, p. 255.
6368:
6353:
6332:
6320:
6308:
6293:
6278:
6266:
6250:
6239:
6227:
6226:
6224:
6221:
6218:
6217:
6198:
6194:
6191:
6185:
6180:
6162:
6123:
6120:
6117:
6106:Brinsmade 1936
6096:
6095:
6092:
6091:
6075:
6074:
6072:
6069:
6067:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6001:
5999:Dihedral angle
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5954:Angle bisector
5951:
5937:
5932:
5926:
5924:
5921:
5918:
5917:
5905:
5893:
5879:
5876:
5873:
5869:
5868:
5856:
5844:
5830:
5827:
5824:
5820:
5819:
5817:
5805:
5791:
5788:
5785:
5781:
5780:
5777:
5774:
5771:
5768:
5763:
5748:In astronomy,
5739:
5738:
5735:
5732:
5725:
5668:vertical angle
5603:
5600:
5590:Leonhard Euler
5562:circular angle
5545:
5542:
5530:
5523:
5517:
5513:
5507:
5503:
5497:
5494:
5490:
5485:
5480:
5474:
5470:
5464:
5460:
5454:
5451:
5447:
5442:
5434:
5430:
5424:
5420:
5414:
5411:
5407:
5400:
5397:
5394:
5391:
5371:
5344:
5341:
5324:
5304:
5301:
5298:
5293:
5288:
5285:
5282:
5279:
5276:
5273:
5270:
5265:
5260:
5257:
5254:
5232:
5208:
5179:
5175:
5171:
5166:
5162:
5158:
5153:
5149:
5146:
5143:
5140:
5137:
5133:
5129:
5125:
5121:
5117:
5113:
5109:
5105:
5101:
5078:
5074:
5070:
5067:
5064:
5044:
5040:
5036:
5033:
5030:
5018:
5015:
5001:
4979:
4958:
4954:
4950:
4947:
4944:
4924:
4920:
4916:
4913:
4910:
4888:
4884:
4880:
4876:
4871:
4867:
4863:
4858:
4854:
4851:
4848:
4845:
4842:
4838:
4834:
4830:
4826:
4822:
4818:
4814:
4810:
4806:
4781:
4777:
4773:
4769:
4764:
4760:
4756:
4752:
4749:
4746:
4743:
4740:
4737:
4733:
4729:
4725:
4721:
4717:
4713:
4709:
4705:
4702:
4674:
4670:
4666:
4662:
4657:
4653:
4649:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4620:
4616:
4612:
4608:
4586:
4583:
4580:
4577:
4574:
4554:
4551:
4543:normal vectors
4528:
4524:
4520:
4516:
4511:
4507:
4503:
4499:
4496:
4493:
4490:
4487:
4484:
4480:
4476:
4472:
4434:
4431:
4427:Pierre Wantzel
4404:
4401:
4354:
4350:
4346:
4334:
4331:
4330:
4329:
4324:, such as the
4318:
4304:
4278:
4275:
4274:
4273:
4230:
4223:
4204:
4201:
4058:Main article:
4055:
4054:Signed angles
4052:
3998:
3978:
3940:
3936:
3933:
3930:
3927:
3924:
3921:
3901:
3898:
3895:
3889:
3886:
3879:
3875:
3871:
3868:
3865:
3859:
3853:
3850:
3843:
3839:
3835:
3832:
3829:
3823:
3817:
3814:
3807:
3803:
3799:
3796:
3793:
3787:
3784:
3781:
3778:
3775:
3772:
3766:
3763:
3757:
3753:
3747:
3741:
3738:
3732:
3728:
3722:
3716:
3713:
3707:
3703:
3697:
3694:
3691:
3688:
3682:
3679:
3676:
3673:
3670:
3667:
3622:
3463:= 1. However,
3419:
3411:
3409:
3406:
3403:
3402:
3391:
3388:
3385:
3381:
3380:
3357:
3354:
3351:
3347:
3346:
3316:
3294:
3291:
3285:
3284:
3273:
3270:
3267:
3263:
3262:
3220:
3217:
3214:
3208:
3207:
3140:
3137:
3134:
3128:
3127:
3101:
3098:
3095:
3088:
3087:
3039:
3036:
3033:
3027:
3026:
2950:
2947:
2939:
2933:
2932:
2885:
2882:
2879:
2873:
2872:
2800:second of time
2796:minute of time
2772:
2769:
2766:
2760:
2759:
2744:
2741:
2738:
2732:
2731:
2685:, also called
2679:
2676:
2673:
2667:
2666:
2580:
2577:
2574:
2568:
2567:
2506:of a degree =
2472:
2469:
2466:
2460:
2459:
2423:
2420:
2417:
2411:
2410:
2367: = 2
2357:
2354:
2346:
2340:
2339:
2336:
2333:
2330:
2258:
2255:
2253:is unaltered.
2226:
2221:
2218:
2213:
2207:
2204:
2200:
2195:
2192:
2119:
2115:
2112:
2109:
2102:
2099:
2094:
2091:
2082:in the angle:
2075:
2071:
2057:
1977:
1974:
1973:
1972:
1968:
1967:to the planes.
1960:dihedral angle
1957:) is called a
1945:
1942:
1941:
1940:
1929:exterior angle
1925:
1922:
1911:
1896:
1864:exterior angle
1859:
1768:interior angle
1763:simple polygon
1749:
1746:
1744:
1743:
1717:
1640:
1630:
1626:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1531:
1528:
1525:
1522:
1519:
1514:
1510:
1506:
1503:
1500:
1495:
1491:
1487:
1484:
1481:
1478:
1475:
1470:
1466:
1462:
1459:
1456:
1451:
1447:
1443:
1441:
1421:of the angle.
1381:turn, 90°, or
1359:
1353:
1349:
1345:
1337:
1333:
1329:
1300:
1297:
1294:
1290:
1287:
1284:
1280:
1277:
1274:
1270:
1267:
1264:
1260:
1257:
1254:
1250:
1247:
1229:
1226:
1193:
1192:
1164:
1044:
1043:
1040:
1037:
999:
975:
966:
963:
962:
959:
956:
953:
950:
947:
946:(0, 100)
944:
941:
934:
933:
930:
927:
924:
921:
918:
917:(0, 90)°
915:
912:
905:
904:
893:
876:
866:
834:
809:
783:
778:
772:
771:
766:
745:
725:
688:
668:
647:
642:
635:
634:
629:
625:
624:
621:
618:
615:
612:
609:
606:
603:
591:
584:
583:
577:
573:
569:
567:
560:
559:
553:
546:
545:
544:
543:
542:
538:
537:
530:
519:complete angle
507:
500:
497:straight angle
473:
466:
401:
390:
372:
369:
360:
357:
319:
315:
312:
309:
256:
252:
248:
244:
240:
228:
225:
149:
146:
132:length to its
26:
9:
6:
4:
3:
2:
8217:
8206:
8203:
8202:
8200:
8189:
8188:
8182:
8181:"Angle"
8177:
8176:
8170:
8167:
8166:
8161:
8157:
8152:
8151:public domain
8136:
8130:
8126:
8121:
8117:
8113:
8109:
8105:
8101:
8097:
8093:
8089:
8084:
8080:
8075:
8064:
8060:
8056:
8055:
8049:
8046:
8042:
8041:
8036:
8031:
8026:
8021:
8017:
8013:
8009:
8004:
8000:
7996:
7992:
7988:
7984:
7980:
7975:
7970:
7967:(5): 053002.
7966:
7962:
7957:
7953:
7949:
7945:
7941:
7937:
7933:
7929:
7925:
7920:
7916:
7915:
7909:
7904:
7899:
7895:
7891:
7886:
7881:
7878:(5): 053001.
7877:
7873:
7869:
7864:
7859:
7854:
7850:
7846:
7841:
7836:
7832:
7828:
7824:
7819:
7815:
7811:
7807:
7803:
7799:
7795:
7791:
7787:
7782:
7778:
7774:
7770:
7766:
7762:
7758:
7754:
7749:
7745:
7741:
7737:
7733:
7729:
7725:
7722:(5): 052001.
7721:
7717:
7712:
7709:
7703:
7699:
7694:
7689:
7685:
7684:
7679:
7674:
7671:
7665:
7661:
7656:
7652:
7651:
7645:
7641:
7637:
7633:
7629:
7625:
7621:
7617:
7612:
7608:
7604:
7600:
7596:
7592:
7588:
7584:
7580:
7575:
7570:
7565:
7561:
7557:
7553:
7549:
7545:
7540:
7536:
7532:
7528:
7524:
7520:
7516:
7511:
7508:
7502:
7498:
7497:
7491:
7490:
7479:, chapter six
7478:
7477:
7472:
7467:
7461:, p. 178
7460:
7456:
7455:Chisholm 1911
7451:
7443:
7437:
7433:
7426:
7410:
7406:
7402:
7396:
7381:
7377:
7371:
7364:
7359:
7352:
7341:
7340:www.boost.org
7337:
7330:
7322:
7318:
7314:
7310:
7306:
7302:
7297:
7292:
7288:
7284:
7277:
7270:
7265:
7258:
7253:
7246:
7241:
7234:
7229:
7227:
7219:
7214:
7212:
7203:
7199:
7195:
7191:
7187:
7183:
7178:
7173:
7169:
7165:
7158:
7156:
7154:
7152:
7144:
7139:
7133:
7129:
7125:
7121:
7117:
7113:
7109:
7105:
7101:
7097:
7093:
7088:
7080:
7076:
7072:
7068:
7064:
7060:
7053:
7045:
7041:
7037:
7033:
7029:
7025:
7018:
7010:
7006:
7002:
6998:
6991:
6983:
6977:
6973:
6972:
6964:
6956:
6952:
6947:
6942:
6938:
6934:
6930:
6926:
6922:
6915:
6908:
6904:
6903:
6895:
6888:
6883:
6871:
6867:
6862:
6857:
6849:
6846:
6827:
6823:of a sector (
6822:
6818:
6814:
6809:
6802:
6797:
6789:
6785:
6780:
6775:
6767:
6763:
6757:
6749:
6745:
6744:
6739:
6733:
6718:
6714:
6710:
6703:
6688:
6684:
6680:
6674:
6660:on 2008-06-28
6659:
6655:
6653:
6648:
6642:
6640:
6625:
6621:
6615:
6597:
6593:
6589:
6582:
6575:
6567:
6563:
6557:
6550:
6549:
6544:
6538:
6529:
6528:
6523:
6520:
6512:
6505:
6498:
6492:
6490:
6488:
6480:
6475:
6467:
6466:
6458:
6452:, p. 97.
6451:
6446:
6432:
6428:
6422:
6416:
6415:Chisholm 1911
6411:
6409:
6394:
6390:
6384:
6377:
6372:
6365:
6360:
6358:
6348:
6347:
6342:
6336:
6329:
6324:
6318:, p. 71.
6317:
6312:
6305:
6300:
6298:
6291:, p. 41.
6290:
6285:
6283:
6276:, p. 18.
6275:
6270:
6263:
6259:
6258:Chisholm 1911
6254:
6248:
6243:
6237:
6232:
6228:
6214:
6192:
6189:
6183:
6168:
6161:
6157:
6153:
6149:
6145:
6141:
6137:
6118:
6107:
6101:
6097:
6087:
6080:
6076:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6034:Phase (waves)
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5985:
5984:Central angle
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5949:
5945:
5941:
5938:
5936:
5933:
5931:
5928:
5927:
5906:
5894:
5880:
5877:
5874:
5871:
5870:
5857:
5845:
5831:
5828:
5825:
5822:
5821:
5818:
5806:
5792:
5789:
5786:
5783:
5782:
5778:
5775:
5772:
5769:
5767:
5764:
5761:
5760:
5757:
5755:
5751:
5746:
5744:
5743:rule of thumb
5736:
5733:
5730:
5729:little finger
5726:
5723:
5719:
5715:
5714:
5713:
5710:
5708:
5704:
5700:
5696:
5695:apparent size
5691:
5689:
5685:
5681:
5677:
5673:
5669:
5664:
5662:
5658:
5654:
5653:
5648:
5647:
5642:
5638:
5633:
5631:
5627:
5623:
5619:
5615:
5614:
5609:
5599:
5597:
5596:
5591:
5587:
5583:
5579:
5575:
5571:
5567:
5563:
5559:
5555:
5551:
5541:
5528:
5521:
5515:
5511:
5505:
5501:
5495:
5492:
5488:
5483:
5478:
5472:
5468:
5462:
5458:
5452:
5449:
5445:
5440:
5432:
5428:
5422:
5418:
5412:
5409:
5405:
5398:
5395:
5392:
5389:
5381:
5379:
5374:
5370:
5366:
5362:
5358:
5354:
5353:metric tensor
5350:
5340:
5338:
5322:
5302:
5299:
5283:
5280:
5277:
5274:
5271:
5255:
5252:
5196:
5195:Hilbert space
5191:
5151:
5144:
5138:
5135:
5131:
5127:
5123:
5111:
5099:
5090:
5065:
5062:
5031:
5028:
5014:
4945:
4942:
4911:
4908:
4899:
4886:
4856:
4849:
4843:
4840:
4836:
4832:
4828:
4816:
4804:
4795:
4792:
4779:
4747:
4741:
4738:
4735:
4731:
4719:
4707:
4703:
4700:
4692:
4690:
4687:In a complex
4685:
4672:
4637:
4631:
4628:
4625:
4614:
4598:
4581:
4578:
4575:
4564:
4560:
4553:Inner product
4550:
4548:
4544:
4539:
4526:
4494:
4488:
4485:
4482:
4474:
4461:
4459:
4455:
4451:
4448:
4444:
4440:
4430:
4428:
4424:
4420:
4414:
4410:
4400:
4399:, biconcave.
4398:
4394:
4390:
4386:
4382:
4372:
4368:
4364:
4344:
4339:
4328:of the angle.
4327:
4323:
4319:
4315:
4311:
4310:
4305:
4302:
4298:
4294:
4293:
4288:
4287:
4286:
4284:
4251:
4247:
4243:
4239:
4238:related angle
4235:
4231:
4228:
4224:
4221:
4217:
4216:
4211:
4207:
4206:
4200:
4197:
4193:
4189:
4184:
4182:
4181:normal vector
4178:
4173:
4171:
4167:
4162:
4160:
4156:
4155:anticlockwise
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4124:
4123:terminal side
4120:
4116:
4112:
4107:
4105:
4101:
4093:
4089:
4085:
4081:
4076:
4071:
4067:
4061:
4048:
4040:
4036:
4030:
4020:
4015:
4010:
3996:
3976:
3968:
3938:
3934:
3931:
3928:
3925:
3922:
3919:
3899:
3896:
3893:
3887:
3884:
3877:
3869:
3866:
3857:
3851:
3848:
3841:
3833:
3830:
3821:
3815:
3812:
3805:
3797:
3794:
3785:
3782:
3779:
3776:
3773:
3770:
3764:
3761:
3755:
3751:
3745:
3739:
3736:
3730:
3726:
3720:
3714:
3711:
3705:
3701:
3695:
3692:
3689:
3686:
3680:
3677:
3674:
3671:
3668:
3665:
3653:
3649:
3648:Taylor series
3644:
3640:
3634:
3630:
3625:
3621:
3614:
3608:
3605:
3599:
3595:
3594:base quantity
3591:
3586:
3584:
3580:
3576:
3573:(rad/s), and
3572:
3568:
3567:angular speed
3564:
3563:angle measure
3559:
3554:
3546:
3542:
3538:
3533:
3512:
3508:
3501:
3491:
3487:
3482:
3478:
3474:
3470:
3457:
3453:
3435:
3431:
3427:
3426:
3416:
3400:
3396:
3392:
3389:
3386:
3383:
3382:
3362:
3361:diameter part
3358:
3355:
3352:
3350:diameter part
3349:
3348:
3325:
3321:
3317:
3295:
3292:
3290:
3287:
3286:
3282:
3278:
3274:
3271:
3268:
3265:
3264:
3240:
3239:
3236:angle of the
3233:
3229:
3225:
3221:
3218:
3215:
3213:
3210:
3209:
3171:
3167:
3166:
3145:
3141:
3138:
3135:
3133:
3130:
3129:
3126:
3122:
3118:
3110:
3105:multiples of
3102:
3099:
3096:
3090:
3089:
3086:
3066:
3062:
3058:
3054:
3050:
3049:
3048:binary radian
3044:
3043:binary degree
3040:
3037:
3034:
3032:
3031:binary degree
3029:
3028:
2972:
2967:
2966:approximately
2963:
2959:
2955:
2951:
2948:
2945:
2940:
2938:
2935:
2934:
2898:
2894:
2890:
2886:
2883:
2880:
2878:
2875:
2874:
2852: turn =
2836: quad =
2801:
2797:
2777:
2773:
2770:
2767:
2765:
2762:
2761:
2757:
2749:
2745:
2742:
2739:
2737:
2734:
2733:
2729:
2725:
2724:triangulation
2721:
2717:
2716:nautical mile
2714:
2710:
2706:
2702:
2698:
2694:
2693:
2688:
2684:
2680:
2677:
2674:
2672:
2669:
2668:
2665:
2629:
2626:of a degree (
2593:
2589:
2585:
2584:second of arc
2581:
2578:
2575:
2573:
2570:
2569:
2565:
2561:
2557:
2556:nautical mile
2489:
2485:
2481:
2477:
2476:minute of arc
2473:
2470:
2467:
2465:
2462:
2461:
2457:
2453:
2449:
2445:
2441:
2436:
2432:
2428:
2424:
2421:
2418:
2416:
2413:
2412:
2408:
2404:
2400:
2374:
2366:
2362:
2358:
2355:
2352:
2347:
2345:
2342:
2341:
2337:
2334:
2331:
2328:
2327:
2324:
2321:
2319:
2315:
2310:
2308:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2279:angular units
2276:
2272:
2263:
2254:
2252:
2248:
2243:
2239:The value of
2237:
2224:
2219:
2216:
2211:
2205:
2202:
2198:
2193:
2190:
2182:
2180:
2176:
2170:
2165:
2148:
2139:
2137:
2133:
2117:
2100:
2097:
2092:
2089:
2081:
2069:
2065:
2060:
2045:
2037:
2027:
2021:
2017:
2015:
2011:
2007:
2003:
1998:
1996:
1992:
1989:
1983:
1969:
1966:
1962:
1961:
1956:
1952:
1948:
1947:
1938:
1934:
1930:
1926:
1923:
1920:
1916:
1915:extended side
1912:
1909:
1905:
1901:
1897:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1865:
1860:
1841:
1837:
1833:
1825:
1821:
1817:
1809:
1808:quadrilateral
1806:
1782:
1778:
1774:
1770:
1769:
1765:is called an
1764:
1760:
1759:
1754:
1742:
1741:of an angle.
1740:
1736:
1730:
1726:
1718:
1712:
1708:
1706:
1703:
1700:
1698:
1697:tangent lines
1693:
1691:
1687:
1686:parallelogram
1683:
1679:
1678:straight line
1675:
1671:
1665:
1641:
1636:
1635:supplementary
1623:
1619:
1617:
1613:
1608:
1606:
1602:
1598:
1593:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1526:
1523:
1520:
1517:
1512:
1508:
1504:
1501:
1498:
1493:
1489:
1482:
1479:
1476:
1473:
1468:
1464:
1460:
1457:
1454:
1449:
1445:
1431:
1427:
1422:
1420:
1415:
1413:
1409:
1403:
1364:
1361:
1360:
1343:
1327:
1326:complementary
1322:
1318:
1315:
1312:
1285:
1282:
1265:
1262:
1245:
1237:
1235:
1225:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1190:
1189:
1184:
1180:
1179:supplementary
1176:
1175:complementary
1172:
1168:
1165:
1161:are adjacent.
1160:
1156:
1151:
1147:
1145:
1141:
1137:
1132:
1128:
1124:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1089:
1083:
1079:
1076:. Both angle
1074:
1068:
1063:
1057:
1053:
1049:
1041:
1038:
1035:
1034:
1033:
1031:
1027:
1023:
1017:
1016:vert. opp. ∠s
1013:
1009:
1005:
1001:
1000:
998:
993:
985:
980:
968:Vertical and
960:
957:
954:
951:
948:
945:
942:
940:
939:
936:
935:
931:
928:
925:
922:
919:
916:
913:
911:
910:
907:
906:
901:
894:
890:
884:
877:
873:
867:
863:
857:
835:
831:
810:
806:
784:
779:
777:
774:
773:
767:
746:
726:
689:
669:
648:
643:
641:
640:
637:
636:
633:
627:
626:
622:
620:reflex angle
619:
616:
614:obtuse angle
613:
610:
607:
604:
601:
600:
588:
564:
556:
550:
541:
535:
534:oblique angle
531:
528:
524:
520:
516:
508:
505:
501:
498:
474:
471:
467:
464:
463:
462:perpendicular
458:
457:
452:
451:
446:
445:
422:
402:
399:
395:
391:
388:
384:
383:
382:
380:
379:
366:
356:
346:
345:
334:
317:
291:
286:
284:
280:
276:
272:
267:
262:
238:
237:Greek letters
234:
224:
222:
218:
217:straight line
214:
210:
206:
202:
200:
196:
192:
188:
183:
173:
169:
164:
159:
155:
145:
143:
139:
135:
131:
127:
123:
122:
117:
113:
108:
106:
102:
98:
97:
92:
88:
87:
82:
81:
76:
75:
71:, called the
70:
66:
62:
55:
51:
46:
40:
33:
19:
8185:
8163:
8140:
8124:
8091:
8087:
8078:
8067:, retrieved
8063:the original
8053:
8038:
8015:
8011:
7964:
7960:
7927:
7923:
7913:
7875:
7871:
7833:(1): 40–47.
7830:
7826:
7789:
7785:
7760:
7756:
7719:
7715:
7697:
7682:
7678:Heath, T. L.
7659:
7649:
7623:
7619:
7582:
7578:
7551:
7547:
7518:
7514:
7495:
7487:Bibliography
7475:
7466:
7459:Heiberg 1908
7450:
7432:Trigonometry
7431:
7425:
7413:. Retrieved
7404:
7395:
7383:. Retrieved
7379:
7370:
7365:, p. 3.
7358:
7350:
7343:. Retrieved
7339:
7329:
7286:
7282:
7276:
7269:Quincey 2021
7264:
7252:
7247:, p. 6.
7240:
7233:Torrens 1986
7218:Quincey 2016
7167:
7163:
7138:
7128:Leonard 2021
7124:Quincey 2021
7104:Torrens 1986
7087:
7065:(2): 84–87.
7062:
7058:
7052:
7027:
7023:
7017:
7000:
6996:
6990:
6970:
6963:
6928:
6924:
6914:
6906:
6901:
6894:
6882:
6869:
6865:
6856:
6847:
6844:
6825:
6820:
6813:Quincey 2016
6808:
6796:
6787:
6783:
6774:
6768:. p. 2.
6765:
6756:
6742:
6732:
6721:. Retrieved
6712:
6702:
6691:. Retrieved
6673:
6662:. Retrieved
6658:the original
6650:
6627:. Retrieved
6623:
6614:
6603:. Retrieved
6591:
6587:
6574:
6547:
6537:
6525:
6515:as cited in
6510:
6504:
6496:
6474:
6464:
6457:
6445:
6434:. Retrieved
6430:
6421:
6396:. Retrieved
6392:
6383:
6371:
6346:The Elements
6345:
6335:
6323:
6311:
6306:, p. 9.
6269:
6262:Heiberg 1908
6253:
6242:
6236:Sidorov 2001
6231:
6167:Quincey 2021
6159:
6151:
6100:
6085:
6079:
6064:Zenith angle
6009:Golden angle
5747:
5740:
5711:
5694:
5692:
5665:
5650:
5644:
5634:
5611:
5605:
5593:
5561:
5560:just as the
5547:
5382:
5377:
5372:
5368:
5364:
5360:
5346:
5192:
5091:
5020:
4900:
4796:
4793:
4693:
4686:
4599:
4562:
4556:
4545:and between
4540:
4462:
4453:
4449:
4445:between two
4442:
4441:, the angle
4436:
4416:
4396:
4392:
4388:
4384:
4380:
4370:
4360:
4342:
4307:
4296:
4290:
4280:
4245:
4241:
4237:
4233:
4226:
4220:right angles
4219:
4213:
4209:
4185:
4174:
4169:
4165:
4163:
4150:
4146:
4142:
4138:
4134:
4126:
4122:
4115:initial side
4114:
4108:
4100:orientations
4097:
4028:
4018:
4014:natural unit
4011:
3967:pure numbers
3654:of an angle
3642:
3638:
3632:
3628:
3619:
3609:
3603:
3587:
3556:In 1993 the
3555:
3544:
3540:
3536:
3510:
3506:
3502:
3480:
3476:
3472:
3455:
3451:
3433:
3429:
3423:
3420:
3401:is 224 zam.
3360:
3319:
3281:Eratosthenes
3276:
3235:
3223:
3190: rad =
3163:
3143:
3123:. See also:
3104:
3068:
3064:
3056:
3052:
3046:
3042:
2965:
2958:scope sights
2943:
2892:
2888:
2871: grad.
2820: rad =
2799:
2795:
2775:
2747:
2719:
2696:
2690:
2686:
2682:
2627:
2591:
2587:
2583:
2563:
2560:great circle
2487:
2483:
2479:
2475:
2455:
2430:
2426:
2398:
2372:
2364:
2360:
2350:
2338:Description
2322:
2311:
2306:
2302:
2278:
2268:
2250:
2246:
2241:
2238:
2183:
2168:
2163:
2146:
2140:
2064:circular arc
2055:
2043:
2035:
2025:
2013:
2001:
1999:
1994:
1990:
1987:
1985:
1958:
1936:
1932:
1928:
1888:
1862:
1839:
1835:
1831:
1823:
1819:
1766:
1738:
1734:
1732:
1728:
1724:
1707:
1704:
1701:
1694:
1681:
1667:
1663:
1634:
1609:
1594:
1429:
1425:
1423:
1418:
1416:
1411:
1408:complementum
1407:
1405:
1362:
1341:
1325:
1316:
1313:
1238:
1233:
1231:
1221:
1217:
1213:
1209:
1205:
1201:
1195:
1186:
1185:angles (see
1183:explementary
1182:
1178:
1174:
1170:
1166:
1158:
1154:
1143:
1139:
1135:
1130:
1126:
1122:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1087:
1081:
1077:
1072:
1066:
1061:
1055:
1051:
1047:
1045:
1021:
1019:
1015:
1011:
1007:
1003:
996:
992:Zenith angle
897:
886:
880:
869:
859:
853:
827:
802:
611:right angle
608:acute angle
592:Reflex angle
539:
533:
526:
522:
518:
514:
504:reflex angle
503:
496:
470:obtuse angle
469:
460:
454:
448:
442:
393:
386:
376:
374:
342:
335:
287:
282:
278:
274:
230:
203:
198:
153:
151:
130:circular arc
119:
115:
109:
94:
86:plane angles
85:
84:
78:
72:
64:
58:
7585:(1): 1–12.
7116:Foster 2010
7096:Romain 1962
6450:Jacobs 1974
6376:Jacobs 1974
6247:Slocum 2007
6156:Foster 2010
6136:Romain 1962
6044:Solid angle
5754:declination
5720:and of the
4458:dot product
4393:amphicoelic
4371:amphicyrtic
4177:orientation
4084:unit circle
4047:Mathematica
4043:plane_angle
3277:hexacontade
3266:hexacontade
3232:sexagesimal
3228:Babylonians
3165:right angle
3121:WP 43S
3085:of a turn.
2937:milliradian
2713:sexagesimal
2554:degrees. A
2435:sexagesimal
2285:( ° ), the
2273:in various
2002:orientation
1877:orientation
1858: turn.
1810:add up to 2
1625:The angles
1198:transversal
984:Hatch marks
974:angle pairs
605:zero angle
572:), obtuse (
555:Right angle
523:round angle
444:right angle
394:acute angle
365:Dutch angle
8094:(1): 1–7.
8088:Metrologia
7974:2108.05704
7961:Metrologia
7924:Metrologia
7885:2203.12392
7872:Metrologia
7827:Metrologia
7786:Metrologia
7716:Metrologia
7620:Metrologia
7579:Metrologia
7296:1705.03765
7283:Metrologia
7177:1604.02373
7164:Metrologia
7120:Mills 2016
6931:(8): 888.
6723:2019-08-05
6693:2019-08-05
6664:2019-08-05
6629:2020-08-31
6605:2019-08-06
6436:2020-08-17
6398:2020-08-17
6316:Moser 1971
6289:Moser 1971
6223:References
6059:Trisection
6039:Protractor
6019:Horn angle
5576:since the
4547:skew lines
4389:sistroidal
4385:xystroidal
4188:navigation
4094:direction.
4064:See also:
3646:, and the
3585:(kg⋅m/s).
3581:(N⋅m) and
3324:Babylonian
3293:144 to 180
2897:navigation
2895:, used in
2776:hour angle
2764:hour angle
2590:, or just
2486:, or just
2452:ballistics
2335:In degrees
1980:See also:
1955:polyhedron
1908:concurrent
1783:add up to
1424:If angles
1419:complement
1342:complement
1138:and angle
1080:and angle
515:full angle
456:orthogonal
387:zero angle
8116:250801509
8045:EMS Press
8018:(3): 97.
7999:236547235
7952:125438811
7840:1409.2794
7814:126032642
7744:234036217
7688:Cambridge
7640:117711734
7607:250750831
7321:119418270
7202:119294905
7100:Eder 1982
6955:225445454
6875:1 rad = 1
6713:HP Museum
6527:MathWorld
6197:Θ
6193:π
6122:⟩
6119:θ
6116:⟨
5703:full moon
5676:elevation
5637:astronomy
5622:longitude
5608:geography
5396:θ
5393:
5284:
5278:≤
5256:
5145:θ
5139:
5120:⟩
5104:⟨
5089:given by
5066:
5032:
4946:
4912:
4850:θ
4844:
4825:⟩
4809:⟨
4748:θ
4742:
4728:⟩
4712:⟨
4704:
4638:θ
4632:
4623:⟩
4607:⟨
4585:⟩
4582:⋅
4576:⋅
4573:⟨
4495:θ
4489:
4475:⋅
4381:cissoidal
4215:congruent
4159:clockwise
4104:rotations
4092:clockwise
4025:1 rad = 1
3935:θ
3929:θ
3926:η
3897:⋯
3870:θ
3867:η
3858:−
3834:θ
3831:η
3798:θ
3795:η
3786:−
3783:θ
3780:η
3774:⋯
3746:−
3696:−
3681:
3672:θ
3669:
3658:becomes:
3569:(rad/s),
3494:1 rad = 1
3488:equal to
3356:≈0.95493°
2854:16
2758:radians.
2728:surveying
2701:kilometre
2588:arcsecond
2576:1,296,000
2572:arcsecond
2484:arcminute
2464:arcminute
2448:astronomy
2212:⋅
2206:π
2191:θ
2090:θ
2068:compasses
1991:congruent
1919:collinear
1904:bisectors
1739:conjugate
1735:explement
1601:cotangent
1574:
1562:
1550:
1538:
1518:
1499:
1474:
1455:
1289:∠
1269:∠
1249:∠
1111:or angle
1069:would be
1058:would be
764:, 1) turn
318:^
261:variables
152:The word
112:magnitude
8199:Category
7698:Geometry
7415:26 April
7409:Archived
6764:(1946).
6740:(1947).
6717:Archived
6687:Archived
6596:Archived
6566:archived
5923:See also
5672:altitude
5670:such as
5618:latitude
5598:(1748).
5554:argument
5359:. Where
5357:tangents
5178:‖
5170:‖
5165:‖
5157:‖
4883:‖
4875:‖
4870:‖
4862:‖
4776:‖
4768:‖
4763:‖
4755:‖
4669:‖
4661:‖
4656:‖
4648:‖
4523:‖
4515:‖
4510:‖
4502:‖
4367:tangents
4297:gradient
4192:bearings
3650:for the
3459:, hence
3437:, where
3353:≈376.991
3328:2
3298:2
3144:quadrant
3132:quadrant
3038:1°33'45"
2962:firearms
2756:𝜏 (tau)
2709:meridian
2271:measured
2179:gradians
2162:, where
1900:triangle
1873:negative
1781:triangle
1670:adjacent
1605:cosecant
1412:complere
1402:triangle
971:adjacent
632:Interval
623:perigon
423:(90° or
142:rotation
8153::
8096:Bibcode
8035:"Angle"
7979:Bibcode
7932:Bibcode
7890:Bibcode
7845:Bibcode
7794:Bibcode
7765:Bibcode
7724:Bibcode
7680:(ed.),
7587:Bibcode
7556:Bibcode
7523:Bibcode
7473:(1892)
7301:Bibcode
7182:Bibcode
7067:Bibcode
7032:Bibcode
6933:Bibcode
6873:, thus
6842:
6830:
6169:), and
5916:minute
5911:⁄
5899:⁄
5887:⁄
5878:0°0′15″
5872:Second
5862:⁄
5850:⁄
5838:⁄
5823:Minute
5811:⁄
5799:⁄
5773:Radians
5770:Degrees
5684:azimuth
5680:horizon
5655:of two
5626:equator
4597:, i.e.
4437:In the
4301:tangent
4265:
4253:
4196:azimuth
4102:and/or
4027:. This
3565:(rad),
3498:rad = 1
3390:≈1.607°
3377:
3365:
3343:
3313:
3255:
3243:
3224:sextant
3212:sextant
3204:
3192:
3188:
3174:
3160:
3148:
3109:radians
3083:
3071:
3023:
3011:
3007:
2992:
2987:
2975:
2949:≈0.057°
2929:
2917:
2913:
2901:
2869:
2850:
2838:
2834:
2822:
2818:
2804:
2792:
2780:
2692:gradian
2660:
2648:
2644:
2632:
2624:
2612:
2608:
2596:
2552:
2540:
2536:
2524:
2520:
2508:
2504:
2492:
2446:and in
2395:
2381:
2356:≈57°17′
2312:In the
2295:history
2291:gradian
2175:degrees
2160:
2143:
2080:radians
2050:radians
2048:
2032:
1885:surface
1879:of the
1856:
1844:
1816:polygon
1801:
1789:
1656:
1644:
1637:angles.
1597:tangent
1397:
1383:
1379:
1367:
1340:is the
1328:angles
1191:below).
1171:adj. ∠s
1153:Angles
1086:180° −
1071:180° −
1060:180° −
1050:equals
851:
839:
825:
813:
800:
788:
762:
750:
741:
729:
721:
709:
705:
693:
684:
672:
664:
652:
568:Acute (
527:perigon
489:
477:
439:
425:
417:
405:
281:,
277:,
209:Proclus
187:English
182:ankylοs
176:ἀγκύλος
168:Cognate
163:angulus
126:measure
105:tangent
52:on the
8147:
8131:
8114:
7997:
7950:
7812:
7742:
7704:
7683:Euclid
7666:
7638:
7605:
7503:
7438:
7385:1 July
7319:
7200:
6978:
6953:
6558:
6341:Euclid
6146:), ◁ (
5901:86,400
5779:Other
5766:Symbol
5697:as an
5572:and a
5552:is an
5351:, the
4644:
4411:, and
4317:lines.
4309:spread
4250:modulo
4131:y-axis
4119:x-axis
4080:x-axis
4068:, and
3912:where
3684:
3592:for a
3579:torque
3322:was a
3320:pechus
3296:2° to
3289:pechus
3241:or is
3094:radian
3069:It is
2884:11.25°
2664:parsec
2592:second
2579:0°0′1″
2517:21,600
2488:minute
2468:21,600
2427:degree
2415:degree
2361:radian
2344:radian
2287:radian
2283:degree
2171:= 360°
2010:spiral
1971:plane.
1965:normal
1951:planes
1917:, are
1902:, the
1805:convex
1674:vertex
1612:prefix
1348:, and
1220:, and
1181:, and
1119:to be
909:degree
776:radian
769:1 turn
723:) turn
666:) turn
645:0 turn
450:normal
419:
205:Euclid
189:word "
134:radius
101:curves
80:vertex
8205:Angle
8160:Angle
8112:S2CID
8069:2 Feb
7995:S2CID
7969:arXiv
7948:S2CID
7880:arXiv
7835:arXiv
7810:S2CID
7740:S2CID
7636:S2CID
7603:S2CID
7345:5 May
7317:S2CID
7291:arXiv
7198:S2CID
7172:arXiv
6951:S2CID
6652:ooPIC
6599:(PDF)
6584:(PDF)
6552:(PDF)
6071:Notes
5889:43200
5867:hour
5852:1,440
5829:0°15′
5784:Hour
5776:Turns
5688:north
5661:Earth
5657:stars
5578:areas
5556:of a
5245:with
5193:in a
4373:(Gr.
4363:curve
4292:slope
4210:equal
4039:Boost
3146:is a
2899:, is
2889:point
2705:centi
2695:, or
2687:grade
2678:0°54′
2594:) is
2490:) is
2275:units
2257:Units
1988:equal
1898:In a
1881:plane
1818:with
1595:(The
932:360°
926:180°
891:) rad
864:) rad
807:) rad
781:0 rad
459:, or
398:sharp
359:Types
199:*ank-
197:root
191:ankle
172:Greek
160:word
158:Latin
154:angle
124:is a
91:plane
74:sides
65:angle
63:, an
32:Angel
8129:ISBN
8071:2010
7702:ISBN
7664:ISBN
7501:ISBN
7436:ISBN
7417:2018
7387:2022
7347:2022
6976:ISBN
6556:ISBN
5944:mean
5904:turn
5855:turn
5816:turn
5762:Unit
5752:and
5722:Moon
5620:and
5363:and
5063:span
5055:and
5029:span
4991:and
4943:span
4935:and
4909:span
4452:and
4417:The
4376:ἀμφί
4349:and
4326:sine
4306:The
4289:The
4232:The
4133:and
3652:sine
3399:turn
3395:turn
3359:The
3318:The
3275:The
3222:The
3142:One
3111:(MUL
3103:The
3100:180°
3061:byte
3053:brad
3041:The
3020:1000
3004:6400
2984:6400
2971:NATO
2960:for
2954:turn
2942:2000
2893:wind
2887:The
2798:and
2748:turn
2746:The
2743:360°
2736:turn
2683:grad
2681:The
2671:grad
2657:3600
2621:3600
2586:(or
2582:The
2543:5.72
2478:(or
2474:The
2471:0°1′
2450:and
2431:turn
2425:The
2384:180°
2359:The
2329:Name
2299:turn
2173:for
2062:, a
2006:turn
1883:(or
1633:are
1629:and
1610:The
1428:and
1332:and
1324:The
1232:The
1157:and
1099:and
961:400
955:200
949:100
920:90°
786:(0,
743:turn
686:turn
650:(0,
639:turn
628:Unit
421:turn
353:∠CAB
349:∠BAC
338:∠BAC
294:∠BAC
290:rays
110:The
69:rays
50:rays
8162:",
8104:doi
8020:doi
8016:66B
7987:doi
7940:doi
7898:doi
7853:doi
7802:doi
7773:doi
7732:doi
7628:doi
7595:doi
7564:doi
7531:doi
7309:doi
7190:doi
7075:doi
7040:doi
7005:doi
6941:doi
6158:),
6150:),
5892:rad
5843:rad
5840:720
5804:rad
5790:15°
5718:Sun
5635:In
5606:In
5592:in
5390:cos
5347:In
5281:dim
5253:dim
5136:cos
4841:cos
4739:cos
4629:cos
4486:cos
4387:or
4353:at
4295:or
4212:or
4194:or
4186:In
4021:= 1
3997:sin
3977:Sin
3963:sin
3959:Sin
3944:rad
3678:sin
3666:Sin
3643:ηrθ
3526:by
3500:.
3475:= 2
3465:rad
3387:224
3384:zam
3345:°.
3219:60°
3139:90°
3117:RPN
3080:256
3055:or
3051:or
3035:256
3025:).
2891:or
2778:is
2771:15°
2754:or
2697:gon
2675:400
2480:MOA
2419:360
2399:rad
2373:rad
2181:):
2029:is
1993:or
1937:not
1727:or
1616:co-
1571:csc
1559:sec
1547:cot
1535:tan
1509:cos
1490:cos
1465:sin
1446:sin
1344:of
1010:or
1006:or
938:gon
914:0°
902:rad
885:, 2
874:rad
832:rad
525:or
400:").
381:):
296:or
231:In
59:In
8201::
8184:,
8110:.
8102:.
8092:22
8090:.
8057:,
8043:,
8037:,
8014:.
8010:.
7993:.
7985:.
7977:.
7965:58
7963:.
7946:.
7938:.
7928:53
7926:.
7896:.
7888:.
7876:59
7874:.
7870:.
7851:.
7843:.
7831:52
7829:.
7825:.
7808:.
7800:.
7790:53
7788:.
7771:.
7761:66
7759:.
7755:.
7738:.
7730:.
7720:58
7718:.
7634:.
7624:47
7622:.
7618:.
7601:.
7593:.
7583:18
7581:.
7562:.
7552:65
7550:.
7546:.
7529:.
7517:.
7457:;
7407:.
7403:.
7378:.
7349:.
7338:.
7315:.
7307:.
7299:.
7287:54
7285:.
7225:^
7210:^
7196:.
7188:.
7180:.
7168:53
7166:.
7150:^
7130:;
7126:;
7122:;
7118:;
7114:;
7110:;
7106:;
7102:;
7098:;
7094:;
7073:.
7063:31
7061:.
7038:.
7028:30
7026:.
7001:30
6999:.
6949:.
6939:.
6929:16
6927:.
6923:.
6877:."
6868:=
6828:=
6786:=
6715:.
6711:.
6681:.
6649:.
6638:^
6622:.
6592:XV
6590:.
6586:.
6564:,
6524:.
6486:^
6429:.
6407:^
6391:.
6356:^
6343:.
6296:^
6281:^
6260:;
6215:).
5946:,
5913:60
5864:60
5813:24
5801:12
5690:.
5548:A
5380:,
5373:ij
5300::=
5272::=
5221:,
4701:Re
4190:,
4161:.
3641:=
3633:rθ
3631:=
3600:,
3539:=
3511:rθ
3509:=
3454:=
3428:=
3374:60
3272:6°
3269:60
3067:.
2910:32
2881:32
2847:24
2815:12
2789:24
2768:24
2730:.
2689:,
2651:30
2646:+
2641:60
2605:60
2549:60
2533:60
2527:30
2501:60
2482:,
2422:1°
2409:.
2407:SI
2320:.
2132:SI
1997:.
1731:.
1666:.
1356:.)
1224:.
1216:,
1212:,
1208:,
1204:,
1196:A
1177:,
1129:=
1024:.
1018:.
943:0
858:,
707:,
521:,
517:,
453:,
255:,
251:,
247:,
243:,
8118:.
8106::
8098::
8028:.
8022::
8001:.
7989::
7981::
7971::
7954:.
7942::
7934::
7906:.
7900::
7892::
7882::
7861:.
7855::
7847::
7837::
7816:.
7804::
7796::
7779:.
7775::
7767::
7746:.
7734::
7726::
7692:.
7642:.
7630::
7609:.
7597::
7589::
7572:.
7566::
7558::
7537:.
7533::
7525::
7519:4
7444:.
7419:.
7389:.
7323:.
7311::
7303::
7293::
7271:.
7235:.
7220:.
7204:.
7192::
7184::
7174::
7145:.
7081:.
7077::
7069::
7046:.
7042::
7034::
7011:.
7007::
6984:.
6957:.
6943::
6935::
6870:r
6866:s
6848:r
6845:θ
6839:2
6836:/
6833:1
6826:A
6821:A
6791:"
6788:r
6784:s
6750:.
6748:7
6726:.
6696:.
6667:.
6632:.
6608:.
6530:.
6439:.
6401:.
6349:.
6211:(
6190:2
6184:=
6179:C
6165:(
6163:C
6160:θ
6154:(
6152:k
6142:(
6140:ם
6134:(
6086:r
5950:)
5942:(
5909:1
5897:1
5884:π
5875:s
5860:1
5848:1
5835:π
5826:m
5809:1
5796:π
5787:h
5674:/
5529:.
5522:|
5516:j
5512:V
5506:i
5502:V
5496:j
5493:i
5489:g
5484:|
5479:|
5473:j
5469:U
5463:i
5459:U
5453:j
5450:i
5446:g
5441:|
5433:j
5429:V
5423:i
5419:U
5413:j
5410:i
5406:g
5399:=
5378:G
5369:g
5365:V
5361:U
5323:k
5303:l
5297:)
5292:W
5287:(
5275:k
5269:)
5264:U
5259:(
5231:W
5207:U
5174:v
5161:u
5152:|
5148:)
5142:(
5132:|
5128:=
5124:|
5116:v
5112:,
5108:u
5100:|
5077:)
5073:v
5069:(
5043:)
5039:u
5035:(
5000:v
4978:u
4957:)
4953:v
4949:(
4923:)
4919:u
4915:(
4887:.
4879:v
4866:u
4857:|
4853:)
4847:(
4837:|
4833:=
4829:|
4821:v
4817:,
4813:u
4805:|
4780:.
4772:v
4759:u
4751:)
4745:(
4736:=
4732:)
4724:v
4720:,
4716:u
4708:(
4673:.
4665:v
4652:u
4641:)
4635:(
4626:=
4619:v
4615:,
4611:u
4579:,
4563:·
4527:.
4519:v
4506:u
4498:)
4492:(
4483:=
4479:v
4471:u
4454:v
4450:u
4443:θ
4357:.
4355:P
4351:B
4347:A
4343:P
4269:π
4262:2
4259:/
4256:1
4246:θ
4242:θ
4229:.
4170:θ
4166:θ
4151:y
4147:x
4139:y
4033:η
4019:η
3939:/
3932:=
3923:=
3920:x
3900:,
3894:+
3888:!
3885:7
3878:7
3874:)
3864:(
3852:!
3849:5
3842:5
3838:)
3828:(
3822:+
3816:!
3813:3
3806:3
3802:)
3792:(
3777:=
3771:+
3765:!
3762:7
3756:7
3752:x
3740:!
3737:5
3731:5
3727:x
3721:+
3715:!
3712:3
3706:3
3702:x
3693:x
3690:=
3687:x
3675:=
3656:θ
3639:s
3629:s
3623:0
3620:ε
3613:η
3604:r
3602:π
3550:ω
3545:r
3543:/
3541:v
3537:ω
3528:θ
3524:r
3520:θ
3516:r
3507:y
3490:1
3481:r
3479:/
3477:A
3473:θ
3456:r
3452:s
3447:r
3443:s
3439:θ
3434:r
3432:/
3430:s
3425:θ
3417:.
3371:/
3368:1
3340:2
3337:/
3334:1
3330:+
3315:°
3310:2
3307:/
3304:1
3300:+
3259:π
3252:6
3249:/
3246:1
3216:6
3201:4
3198:/
3195:1
3185:2
3182:/
3178:π
3157:4
3154:/
3151:1
3136:4
3113:π
3107:π
3097:2
3092:π
3077:/
3074:1
3065:n
3017:/
3014:1
3001:/
2997:π
2995:2
2990:(
2981:/
2978:1
2969:"
2944:π
2926:8
2923:/
2920:1
2907:/
2904:1
2866:3
2863:/
2860:2
2856:+
2844:/
2841:1
2831:6
2828:/
2825:1
2812:/
2808:π
2786:/
2783:1
2752:π
2740:1
2720:n
2718:(
2654:/
2638:/
2635:7
2628:n
2618:/
2615:1
2602:/
2599:1
2564:n
2546:/
2530:/
2514:/
2511:1
2498:/
2495:1
2456:n
2454:(
2391:π
2387:/
2377:π
2369:π
2365:n
2351:π
2349:2
2307:n
2303:n
2251:r
2249:/
2247:s
2242:θ
2225:.
2220:r
2217:s
2203:2
2199:k
2194:=
2169:k
2164:k
2156:π
2154:2
2151:/
2147:k
2118:.
2114:d
2111:a
2108:r
2101:r
2098:s
2093:=
2076:r
2072:s
2059:θ
2052:.
2044:r
2040:/
2036:s
2026:θ
1935:(
1921:.
1853:2
1850:/
1847:1
1840:n
1836:n
1832:n
1828:π
1824:n
1820:n
1812:π
1798:2
1795:/
1792:1
1785:π
1721:π
1660:π
1653:2
1650:/
1647:1
1631:b
1627:a
1614:"
1577:B
1568:=
1565:A
1553:B
1544:=
1541:A
1527:1
1524:=
1521:B
1513:2
1505:+
1502:A
1494:2
1483:1
1480:=
1477:B
1469:2
1461:+
1458:A
1450:2
1430:B
1426:A
1394:2
1391:/
1387:π
1376:4
1373:/
1370:1
1354:b
1350:a
1346:a
1338:b
1336:(
1334:b
1330:a
1299:C
1296:O
1293:B
1286:m
1283:+
1279:B
1276:O
1273:A
1266:m
1263:=
1259:C
1256:O
1253:A
1246:m
1159:B
1155:A
1144:x
1140:B
1136:A
1131:x
1127:x
1123:x
1117:B
1113:D
1109:C
1105:B
1101:D
1097:C
1093:B
1088:x
1082:D
1078:C
1073:x
1067:D
1062:x
1056:C
1052:x
1048:A
994:.
899:π
896:2
888:π
882:π
879:(
871:π
861:π
855:π
848:2
845:/
842:1
837:(
829:π
822:2
819:/
816:1
804:π
797:2
794:/
791:1
759:2
756:/
753:1
748:(
738:2
735:/
732:1
718:2
715:/
712:1
702:4
699:/
696:1
691:(
681:4
678:/
675:1
661:4
658:/
655:1
578:c
574:b
570:a
536:.
529:.
511:π
506:.
499:.
493:π
486:2
483:/
480:1
465:.
436:2
433:/
429:π
414:4
411:/
408:1
389:.
367:.
314:C
311:A
308:B
283:c
279:b
275:a
266:π
257:φ
253:θ
249:γ
245:β
241:α
239:(
179:(
41:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.