748:
2789:
2083:
5078:
2813:
1532:
1778:
693:
2187:
49:
1646:
6836:
5178:
980:(mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used.
7057:
5249:
6739:
5664:
6249:
5553:
1691:
6991:
5731:
5349:
5482:
7144:
6920:
7207:
6609:
5122:
7335:
6690:
5872:
5799:
5611:
3537:
4770:
5010:
1499:
7489:
Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a
3938:
2671:
3647:
1673:, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:
6440:
3329:
7524:; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its
1990:
7325:
4642:
6242:
4908:
6684:
6165:
6035:
6830:
3783:
6351:
4016:
1817:, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is
5476:
6093:
5793:
5243:
3314:
6564:
5964:
4900:
4614:
3157:
5547:
7138:
5343:
4825:
3808:
3024:
356:. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:
338:
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.
7626:
7668:
5866:
7589:
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.
2365:
7201:
5725:
3238:
1133:
is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician
5405:
2899:
2483:
2433:
2231:
335:, though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
6914:
5172:
4480:
5605:
2765:
1750:
7424:
4412:
2274:
1369:
140:
4066:
4106:
7051:
6985:
1342:
8369:
6733:
5658:
4847:
3093:
4257:
9590:
8945:
4561:
4144:
2506:
2066:
5072:
6284:
4291:
4206:
4170:
6598:
4500:
4356:
4314:
1858:
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The
8500:
5116:
3053:
7494:
drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the
6490:
6463:
5902:
4634:
2713:. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use
8982:
9680:
3552:
3532:{\displaystyle \oint _{t_{0}}^{t_{1}}x{\dot {y}}\,dt=-\oint _{t_{0}}^{t_{1}}y{\dot {x}}\,dt={1 \over 2}\oint _{t_{0}}^{t_{1}}(x{\dot {y}}-y{\dot {x}})\,dt}
6356:
761:
and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:
4765:{\displaystyle A=\iint _{D}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}+1}}\,dx\,dy.}
192:. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of
8529:
1894:
7221:
3677:
6170:
5005:{\displaystyle A=\iint _{D}\left|{\frac {\partial \mathbf {r} }{\partial u}}\times {\frac {\partial \mathbf {r} }{\partial v}}\right|\,du\,dv.}
2169:
is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to
1023:
was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his
6623:
6098:
5970:
2503:
are its included sides. If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of
8733:
8296:
8703:
8576:
8975:
8153:
1093:, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with
285:
is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the
8332:
However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a circle:
7858:
1264:
in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an
8447:
6753:
6294:
3955:
1177:
1089:, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular
8968:
8796:
8727:
8697:
8625:
8570:
8543:
8325:
8290:
8031:
7939:
7909:
8863:
8802:
5418:
7971:
7829:
6040:
5746:
5192:
3243:
2121:) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a
708:
has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in
8493:
8227:
6508:
5910:
4852:
4566:
3106:
1603:
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a
5505:
1172:
A formula equivalent to Heron's was discovered by the
Chinese independently of the Greeks. It was published in 1247 in
7993:
7075:
5271:
8093:
8072:
8051:
4782:
2948:
2816:
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions
8763:
7786:
9716:
9691:
9652:
7862:
7696:
7595:
7575:(one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths.
764:
3 metres × 2 metres = 6 m. This is equivalent to 6 million square millimetres. Other useful conversions are:
7634:
5814:
1217:
1061:
whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book
17:
2301:
7158:
5678:
3169:
1086:
155:
5354:
2844:
2438:
2382:
2193:
1880:
was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to
8994:
8912:
8177:
1137:
proved that the area of a triangle is half that of a parallelogram with the same base and height in his book
228:
7738:
7507:
6864:
5136:
1494:{\displaystyle A={\frac {1}{2}}{\Biggl \vert }\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i}){\Biggr \vert }}
4435:
1296:
1233:
1039:, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.
1032:
5571:
3933:{\displaystyle A={\frac {(b^{2}-4ac)^{3/2}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha )^{3},\qquad a\neq 0.}
3654:
2731:
1716:
7721:
7387:
4379:
2244:
2142:
1221:
239:
long. A shape with an area of three square metres would have the same area as three such squares. In
114:
107:
4029:
9623:
9428:
9282:
7701:
7679:
2829:(b is defined as the larger of the two values) on the horizontal axis, is given by the integral from
8842:
4071:
2125:, the side surface can be flattened out into a sector of a circle, and the resulting area computed.
621:. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e.
9395:
7579:
7363:
7009:
6934:
1301:
1213:
1198:
1024:
8335:
8203:
7727:
7582:
for triangles states that the triangle of greatest area among all those with a given perimeter is
7538:
through the center bisect the area. In the case of a circle they are the diameters of the circle.
6704:
5625:
4830:
3063:
2666:{\displaystyle {\tfrac {1}{2}}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})}
2136:, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by
9406:
7558:
4221:
1572:
That is, the area of the rectangle is the length multiplied by the width. As a special case, as
1063:
9566:
9466:
4519:
The general formula for the surface area of the graph of a continuously differentiable function
8837:
7516:
of the triangle (which connect the sides' midpoints with the opposite vertices), and these are
4522:
4119:
2918:
2910:
2677:
and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points
2036:
1107:
894:
305:
248:
185:
8717:
8535:
8280:
8088:. Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra. pp. 58–59.
2821:
The area between a positive-valued curve and the horizontal axis, measured between two values
311:
Area plays an important role in modern mathematics. In addition to its obvious importance in
208:
of a solid (a three-dimensional concept). Two different regions may have the same area (as in
9541:
9411:
9149:
8687:
8560:
7901:
7889:
7351:
7214:
5044:
1703:
1654:
1094:
928:
747:
324:
278:
196:
necessary to cover the surface with a single coat. It is the two-dimensional analogue of the
159:
151:
31:
8467:
8317:
8311:
8144:
7883:
6257:
4270:
4181:
4149:
2097:
of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a
9863:
9788:
9416:
9188:
8859:
8652:
7512:
There are an infinitude of lines that bisect the area of a triangle. Three of them are the
6571:
5496:
4485:
4329:
4296:
1877:
1202:
1043:
1020:
1014:
5092:
3029:
1640:
8:
9594:
9323:
9271:
8991:
8276:
7925:
7568:
The circle has the largest area of any two-dimensional object having the same perimeter.
6998:
3940:
The above remains valid if one of the bounding functions is linear instead of quadratic.
3802:
2906:
2114:
2110:
2098:
1859:
1114:
1028:
286:
209:
8422:
8252:
6472:
6445:
3642:{\displaystyle {1 \over 2}\oint _{t_{0}}^{t_{1}}{\vec {u}}\times {\dot {\vec {u}}}\,dt.}
2435:
is half of its perimeter. If an angle and its two included sides are given, the area is
1121:
for the area of a triangle in terms of its sides, and a proof can be found in his book,
8937:
8929:
8669:
8617:
8111:
7711:
5887:
5411:
5262:
4776:
4619:
4373:
3671:
2294:
2133:
2094:
1804:
1517:
1157:
1118:
1054:
54:
7481:(where the apothem is the distance from the center to the nearest point on any side).
9613:
9531:
9367:
9126:
8907:
8792:
8759:
8752:
8723:
8693:
8673:
8566:
8539:
8398:
8321:
8286:
8089:
8068:
8047:
8027:
8019:
7967:
7960:
7935:
7929:
7905:
7782:
7775:
7690:
7562:
7513:
7495:
6856:
6835:
4429:
3096:
2776:
1885:
1881:
1356:
1261:
1245:
1036:
301:
8941:
8178:"Land Measurement Units in India: Standard Measurement Units, Land Conversion Table"
1237:
9700:
9670:
9636:
9355:
9311:
9266:
8925:
8921:
8883:. R. Honsberger (ed.). Washington, DC: Mathematical Association of America, p. 147.
8847:
8661:
8609:
8468:"Bridge Named After the Mathematician Who Discovered the Chinese Remainder Theorem"
8401:
7743:
7535:
7517:
6435:{\displaystyle {\tfrac {a}{2}}=r\tan({\tfrac {\pi }{n}})=R\sin({\tfrac {\pi }{n}})}
4323:
3164:
2674:
2118:
1794:
1360:
1241:
1209:
1161:
969:
The barn is commonly used in describing the cross-sectional area of interaction in
796:
729:
713:
332:
8594:
7433:. Thus a circle has the largest area of any closed figure with a given perimeter.
7350:
The areas of irregular (and thus arbitrary) polygons can be calculated using the "
5077:
2288:
lies to the other vertex of the triangle). This formula can be used if the height
1706:
triangles, as shown in the figure to the right. It follows that the area of each
1074:, and the area of a triangle is half the base times the height, yielding the area
9945:
9905:
8855:
8825:
8786:
8525:
8449:
The
Aryabhatiya of Aryabhata: An Ancient Indian Work on Mathematics and Astronomy
7897:
7682:, a cyclic quadrilateral with integer sides, integer diagonals, and integer area.
7631:
The ratio of the area to the square of the perimeter of an equilateral triangle,
7547:
7474:
6616:
5879:
5564:
2788:
2122:
2082:
2015:
2011:
1814:
1782:
970:
755:
Calculation of the area of a square whose length and width are 1 metre would be:
736:
9708:
9644:
7963:
7825:
1164:, expressed the area of a triangle as one-half the base times the height in the
1049:
dealt with equality of areas between two-dimensional figures. The mathematician
9889:
9853:
9778:
9559:
9446:
9336:
9237:
8960:
7572:
7534:
All area bisectors of a circle or other ellipse go through the center, and any
2721:
1670:
1620:
1292:
1058:
776:
705:
347:
320:
217:
189:
8665:
7592:
The ratio of the area of the incircle to the area of an equilateral triangle,
2792:
Integration can be thought of as measuring the area under a curve, defined by
304:, but computing the surface area of a more complicated shape usually requires
9982:
9976:
9763:
9631:
9345:
8851:
7885:
7685:
7546:
Given a wire contour, the surface of least area spanning ("filling") it is a
7452:
7448:
are arbitrarily close to 0° and the other two are arbitrarily close to 180°.
5618:
3317:
1985:{\displaystyle A\;=\;2\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,dx\;=\;\pi r^{2}.}
1824:
1786:
1662:
1636:
1206:
1149:
936:
904:
247:
is defined to have area one, and the area of any other shape or surface is a
171:
7320:{\displaystyle A=2\pi \int _{a}^{b}\!f(x){\sqrt {1+\left^{2}}}\mathrm {d} x}
9838:
9824:
9724:
9660:
9385:
9168:
9079:
6846:
6492:
2106:
2077:
1269:
783:
709:
297:
232:
181:
176:
92:
84:
9206:
8755:
8142:
7778:
7746:, a cyclic pentagon whose side lengths and area are all rational numbers.
7583:
7551:
6237:{\displaystyle \quad ={\tfrac {1}{2}}nR^{2}\sin({\tfrac {2\pi }{n}})\,\!}
3801:) is the quadratic lower bound. By the area integral formulas above and
2930:
1624:
1194:
1181:
1166:
958:
804:
728:(mi), and so forth. Algebraically, these units can be thought of as the
725:
721:
717:
316:
251:
244:
240:
201:
8933:
7854:
7704:, an inner triangle with one-seventh the area of the reference triangle.
7528:). There are either one, two, or three of these for any given triangle.
2186:
8621:
7732:
3658:
2812:
2137:
2086:
1616:
1604:
1531:
1212:
in a circle) in terms of its sides. In 1842, the German mathematicians
1153:
1126:
1050:
977:
900:
828:
300:. Formulas for the surface areas of simple shapes were computed by the
213:
8650:
Trainin, J. (November 2007). "An elementary proof of Pick's theorem".
2001:
1884:. Using modern methods, the area of a circle can be computed using a
1286:
1265:
48:
9299:
9226:
9214:
8406:
7371:
6679:{\displaystyle A={\frac {\theta }{2}}r^{2}={\frac {L\cdot r}{2}}\,\!}
6287:
5671:
5085:
1803:(more properly called the area enclosed by a circle or the area of a
1757:
1666:
1544:
1145:
692:
266:
8613:
6160:{\displaystyle \quad ={\tfrac {1}{4n}}p^{2}\cot({\tfrac {\pi }{n}})}
6030:{\displaystyle \quad ={\tfrac {1}{4}}na^{2}\cot({\tfrac {\pi }{n}})}
1645:
923:
1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres
795:
In non-metric units, the conversion between two square units is the
296:, cone, or cylinder, the area of its boundary surface is called the
9603:
9502:
9256:
9115:
9069:
7531:
Any line through the midpoint of a parallelogram bisects the area.
7525:
7521:
6746:
6600:
6465:
5490:
5256:
5185:
5129:
4215:
2914:
2728:
coordinates) such that all the polygon's vertices are grid points:
2724:
constructed on a grid of equal-distanced points (i.e., points with
2714:
2238:
2170:
1873:
1707:
1699:
1658:
1612:
328:
312:
282:
262:
5177:
3674:, we first subtract one from the other, writing the difference as
2109:
can be obtained by cutting surfaces and flattening them out (see:
1649:
A parallelogram can be cut up and re-arranged to form a rectangle.
235:(written as m), which is the area of a square whose sides are one
30:
This article is about the geometric quantity. For other uses, see
9929:
7491:
7478:
7441:
7343:
The above calculations show how to find the areas of many common
7056:
6738:
6697:
5807:
5739:
5663:
5560:
5248:
2725:
2007:
1761:
1090:
916:
769:
697:
274:
258:
79:
8879:
Chakerian, G.D. (1979) "A Distorted View of
Geometry." Ch. 7 in
6248:
5552:
735:
The SI unit of area is the square metre, which is considered an
9843:
9812:
9768:
9746:
9200:
9099:
9064:
7456:
7430:
7344:
6990:
6927:
6501:
5730:
5481:
5348:
5037:
3665:
2129:
2090:
1872:
This argument is actually a simple application of the ideas of
1800:
1690:
1582:
in the case of a square, the area of a square with side length
1134:
932:
834:
1 square mile = 3,097,600 square yards = 27,878,400 square feet
293:
270:
224:
205:
197:
7206:
7143:
6919:
6608:
5121:
1657:. This involves cutting a shape into pieces, whose areas must
1653:
Most other simple formulas for area follow from the method of
223:
The area of a shape can be measured by comparing the shape to
9877:
9802:
9736:
9340:
9246:
9241:
9178:
9153:
8562:
Calculus for
Scientists and Engineers: An Analytical Approach
7445:
7334:
7151:
6825:{\displaystyle A=\int _{a}^{b}f(x)\mathrm {d} x,\ f(x)\geq 0}
6689:
1777:
1608:
1543:
The most basic area formula is the formula for the area of a
1249:
684:
It can be proved that such an area function actually exists.
353:
236:
193:
167:
100:
8396:
8204:"Land is measured in what units in India: All Types In 2023"
5871:
5798:
4775:
An even more general formula for the area of the graph of a
3778:{\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha )(x-\beta )}
1756:
Similar arguments can be used to find area formulas for the
1630:
1057:
to show that the area inside a circle is equal to that of a
216:, "area" sometimes is used to refer to the region, as in a "
9959:
9919:
9492:
9089:
9054:
8228:"Standard Land Measurement Units in India - Times Property"
7064:
5610:
4175:
3949:
2292:
is known. If the lengths of the three sides are known then
2010:
is related to the formula of a circle; for an ellipse with
943:
808:
751:
Although there are 10 mm in 1 cm, there are 100 mm in 1 cm.
352:
An approach to defining what is meant by "area" is through
7440:
could have an arbitrarily small area, as illustrated by a
6346:{\displaystyle r={\tfrac {a}{2}}\cot({\tfrac {\pi }{n}}),}
2775:
is the number of boundary points. This result is known as
2132:
is more difficult to derive: because a sphere has nonzero
1862:
of the areas of the approximate parallelograms is exactly
1807:) is based on a similar method. Given a circle of radius
983:
Some traditional South Asian units that have fixed value:
815:
the relationship between square feet and square inches is
799:
of the conversion between the corresponding length units.
7455:(the term for the perimeter of a circle) equals half the
7444:
that is "tipped over" arbitrarily far so that two of its
4011:{\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}
8253:"9 Land Measurement Units in India You Must Know - 2022"
7931:
A History of the
Calculus and Its Conceptual Development
7735:, an instrument for measuring small areas, e.g. on maps.
3943:
1698:
However, the same parallelogram can also be cut along a
8065:
Real
Analysis: modern techniques and their applications
7628:, is larger than that of any non-equilateral triangle.
7508:
Bisection § Area bisectors and perimeter bisectors
1661:
to the area of the original shape. For an example, any
1129:
knew the formula over two centuries earlier, and since
712:(m), square centimetres (cm), square millimetres (mm),
8423:"Euclid's Proof of the Pythagorean Theorem | Synaptic"
7955:
7953:
7951:
7855:"Resolution 12 of the 11th meeting of the CGPM (1960)"
6418:
6388:
6361:
6326:
6305:
6213:
6179:
6143:
6107:
6071:
6013:
5979:
5471:{\displaystyle A={\frac {c}{4}}{\sqrt {4a^{2}-c^{2}}}}
5365:
4636:
is a region in the xy-plane with the smooth boundary:
2511:
2443:
2393:
2249:
2204:
9569:
8338:
7637:
7598:
7390:
7224:
7161:
7078:
7012:
6937:
6867:
6756:
6707:
6626:
6574:
6511:
6475:
6448:
6359:
6297:
6260:
6173:
6101:
6088:{\displaystyle \quad =nr^{2}\tan({\tfrac {\pi }{n}})}
6043:
5973:
5913:
5890:
5817:
5749:
5681:
5628:
5574:
5508:
5421:
5357:
5274:
5195:
5139:
5095:
5047:
4911:
4855:
4833:
4785:
4645:
4622:
4569:
4525:
4488:
4438:
4382:
4332:
4299:
4273:
4224:
4184:
4152:
4122:
4074:
4032:
3958:
3811:
3680:
3555:
3332:
3246:
3172:
3109:
3066:
3032:
2951:
2847:
2734:
2509:
2441:
2385:
2304:
2247:
2196:
2039:
1897:
1719:
1372:
1304:
331:, the area of a subset of the plane is defined using
117:
7714:, a generalization of the one-seventh area triangle.
7436:
At the other extreme, a figure with given perimeter
5788:{\displaystyle A={\frac {3}{2}}{\sqrt {3}}a^{2}\,\!}
4849:
is a continuously differentiable vector function of
2771:
is the number of grid points inside the polygon and
1081:
for the disk.) Archimedes approximated the value of
899:
There are several other common units for area. The
8905:
7948:
5238:{\displaystyle A={\frac {1}{2}}ab\sin(\gamma )\,\!}
4514:
3309:{\displaystyle {\vec {u}}(t_{0})={\vec {u}}(t_{1})}
1240:in the 17th century allowed the development of the
1125:, written around 60 CE. It has been suggested that
946:is also commonly used to measure land areas, where
888:
315:and calculus, area is related to the definition of
9584:
8363:
8143:Bureau international des poids et mesures (2006).
8110:
7662:
7620:
7429:and equality holds if and only if the curve is a
7418:
7319:
7195:
7132:
7045:
6979:
6908:
6824:
6727:
6678:
6592:
6559:{\displaystyle A=\pi r^{2}={\frac {\pi d^{2}}{4}}}
6558:
6484:
6457:
6434:
6345:
6278:
6236:
6159:
6087:
6029:
5959:{\displaystyle A=n{\frac {ar}{2}}={\frac {pr}{2}}}
5958:
5896:
5860:
5787:
5719:
5652:
5599:
5541:
5470:
5399:
5337:
5237:
5166:
5110:
5066:
5004:
4895:{\displaystyle (u,v)\in D\subset \mathbb {R} ^{2}}
4894:
4841:
4819:
4764:
4628:
4609:{\displaystyle (x,y)\in D\subset \mathbb {R} ^{2}}
4608:
4555:
4494:
4474:
4406:
4350:
4308:
4285:
4251:
4200:
4164:
4138:
4100:
4060:
4010:
3932:
3777:
3641:
3531:
3308:
3232:
3152:{\displaystyle A={1 \over 2}\int r^{2}\,d\theta .}
3151:
3087:
3047:
3018:
2893:
2759:
2665:
2477:
2427:
2359:
2268:
2225:
2060:
1984:
1744:
1493:
1336:
1255:
1085:(and hence the area of a unit-radius circle) with
957:On the atomic scale, area is measured in units of
134:
8316:(5th. ed.). Toronto ON: Brook/Cole. p.
7693:, a triangle with integer sides and integer area.
7252:
6724:
6675:
6233:
5857:
5784:
5716:
5649:
5542:{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}\,\!}
5538:
5334:
5234:
5163:
4097:
2181:
1486:
1391:
950:1 acre = 4,840 square yards = 43,560 square feet.
779:square centimetres = 1,000,000 square millimetres
9974:
8990:
8747:
8745:
8743:
8531:Problem Solving Through Recreational Mathematics
8494:"Calculating The Area And Centroid Of A Polygon"
8381:
8285:. Courier Dover Publications. pp. 121–132.
7357:
7133:{\displaystyle A=\pi r(r+{\sqrt {r^{2}+h^{2}}})}
5338:{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}\,\!}
927:Other uncommon metric units of area include the
8113:Elementary Geometry from an Advanced Standpoint
4820:{\displaystyle \mathbf {r} =\mathbf {r} (u,v),}
3019:{\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,}
8778:
8751:
8313:Single variable calculus early transcendentals
8138:
8136:
8134:
8132:
7959:
7774:
1813:, it is possible to partition the circle into
1694:A parallelogram split into two equal triangles
8976:
8740:
8524:
8250:
4026:is the height. That can also be rewritten as
1272:of various curved three-dimensional objects.
8908:"Triangles, ellipses, and cubic polynomials"
8894:100 Great Problems of Elementary Mathematics
8875:
8873:
8722:. PT Grafindo Media Pratama. pp. 157–.
8475:Notices of the American Mathematical Society
8386:. Oxford University Press. pp. 321–323.
8024:Differential Geometry of Curves and Surfaces
7670:is larger than that for any other triangle.
7621:{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}
3666:Bounded area between two quadratic functions
8692:. PT Grafindo Media Pratama. pp. 51–.
8588:
8586:
8129:
7770:
7768:
7766:
7764:
7762:
7760:
7663:{\displaystyle {\frac {1}{12{\sqrt {3}}}},}
7451:For a circle, the ratio of the area to the
5861:{\displaystyle A=2(1+{\sqrt {2}})a^{2}\,\!}
954:An acre is approximately 40% of a hectare.
613:be a set enclosed between two step regions
8983:
8969:
8784:
8455:. University of Chicago Press. p. 26.
7366:states that, for a closed curve of length
2837:of the function that represents the curve:
2360:{\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}
2113:). For example, if the side surface of a
2071:
1965:
1961:
1905:
1901:
915:Though the are has fallen out of use, the
47:
8870:
8841:
8015:
8013:
7859:Bureau International des Poids et Mesures
7462:. This can be seen from the area formula
7196:{\displaystyle A=4\pi ^{2}\cdot R\cdot r}
6723:
6674:
6232:
5856:
5783:
5720:{\displaystyle A={\frac {(a+c)h}{2}}\,\!}
5715:
5648:
5537:
5333:
5233:
5162:
4992:
4985:
4882:
4752:
4745:
4596:
4096:
3629:
3522:
3434:
3377:
3233:{\displaystyle {\vec {u}}(t)=(x(t),y(t))}
3139:
3006:
2881:
2093:is exactly four times the area of a flat
1954:
1834:. Thus, the total area of the circle is
1631:Dissection, parallelograms, and triangles
842:1 square inch = 6.4516 square centimetres
838:In addition, conversion factors include:
323:, and is a basic property of surfaces in
273:. Using these formulas, the area of any
8583:
7757:
5400:{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
4172:is the lateral surface area of the cone.
4022:is the radius of the circular base, and
2894:{\displaystyle A=\int _{a}^{b}f(x)\,dx.}
2811:
2787:
2478:{\displaystyle {\tfrac {1}{2}}ab\sin(C)}
2428:{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
2226:{\displaystyle A={\tfrac {b\cdot h}{2}}}
2185:
2081:
2006:The formula for the area enclosed by an
1776:
1767:
1689:
1644:
1530:
1220:independently found a formula, known as
919:is still commonly used to measure land:
746:
691:
8649:
8384:A History of Greek Mathematics (Vol II)
8309:
8083:
8077:
7728:Derivation of the formula of a pentagon
3655:Green's theorem § Area calculation
2284:is the distance from the line on which
1785:which rearrange to form an approximate
1710:is half the area of the parallelogram:
1244:for the area of any polygon with known
1227:
993:1 Kanal = 0.125 acre (1 acre = 8 kanal)
261:for the areas of simple shapes such as
231:(SI), the standard unit of area is the
14:
9975:
8906:Minda, D.; Phelps, S. (October 2008).
8823:
8592:
8491:
8201:
8146:The International System of Units (SI)
8067:, John Wiley & Sons, Inc., p. 20,
8010:
6909:{\displaystyle A=4\pi r^{2}=\pi d^{2}}
5167:{\displaystyle A={\frac {1}{2}}bh\,\!}
3055:is the curve with the greater y-value.
2128:The formula for the surface area of a
1178:Mathematical Treatise in Nine Sections
823:where 144 = 12 = 12 × 12. Similarly:
121:
8964:
8558:
8445:
8397:
8275:
8269:
8108:
8102:
7924:
5021:Additional common formulas for area:
3944:Surface area of 3-dimensional figures
3670:To find the bounded area between two
1619:, this formula can be used to define
1224:, for the area of any quadrilateral.
1188:
903:was the original unit of area in the
8643:
8465:
8036:
7820:
7818:
7816:
7814:
7812:
7810:
7808:
7806:
7804:
2933:the integral of the other function,
2673:. This formula is also known as the
1359:are known, the area is given by the
341:
281:. For shapes with curved boundary,
7877:
7849:
7847:
5015:
4475:{\displaystyle 2(\ell w+\ell h+wh)}
3793:) is the quadratic upper bound and
2783:
2379:are the sides of the triangle, and
2176:
1869:, which is the area of the circle.
1535:The area of this rectangle is
1280:
790:
732:of the corresponding length units.
279:dividing the polygon into triangles
204:(a one-dimensional concept) or the
24:
8805:from the original on 20 March 2017
7890:"Chapter 3: Polygon Triangulation"
7310:
6791:
6247:
5600:{\displaystyle A={\frac {1}{2}}de}
4971:
4961:
4946:
4936:
4718:
4710:
4680:
4672:
4422:is the perimeter of the base, and
2760:{\displaystyle i+{\frac {b}{2}}-1}
2089:showed that the surface area of a
1745:{\displaystyle A={\frac {1}{2}}bh}
1197:developed a formula, now known as
819:1 square foot = 144 square inches,
287:historical development of calculus
53:The areas of this square and this
25:
9994:
8956:
8766:from the original on 21 June 2012
8631:from the original on 27 June 2012
8466:Xu, Wenwen; Yu, Ning (May 2013).
7974:from the original on 23 June 2012
7801:
7501:
7419:{\displaystyle 4\pi A\leq L^{2},}
4407:{\displaystyle B+{\frac {PL}{2}}}
4116:is the slant height of the cone.
2269:{\displaystyle {\tfrac {1}{2}}Bh}
1641:Parallelogram § Area formula
135:{\displaystyle {\mathsf {L}}^{2}}
27:Size of a two-dimensional surface
8948:from the original on 2016-11-04.
8866:from the original on 2014-04-08.
8830:Journal of Differential Geometry
8736:from the original on 2016-12-23.
8706:from the original on 2017-03-20.
8579:from the original on 2016-05-05.
8299:from the original on 2016-05-01.
7884:Mark de Berg; Marc van Kreveld;
7844:
7832:from the original on 2 July 2012
7477:is half its perimeter times the
7381:of the region that it encloses,
7333:
7205:
7142:
7055:
6989:
6918:
6834:
6737:
6688:
6607:
5870:
5797:
5729:
5662:
5609:
5551:
5480:
5347:
5247:
5176:
5120:
5076:
4965:
4940:
4835:
4795:
4787:
4515:General formula for surface area
4366:is the perimeter of a base, and
4061:{\displaystyle \pi r^{2}+\pi rl}
3657:.) This is the principle of the
1547:. Given a rectangle with length
1275:
1106:This section is an excerpt from
1100:
889:Other units including historical
162:. The area of a plane region or
9007:Linear/translational quantities
8899:
8886:
8817:
8719:Get Success UN +SPMB Matematika
8710:
8680:
8602:The College Mathematics Journal
8593:Braden, Bart (September 1986).
8552:
8518:
8506:from the original on 2012-09-16
8485:
8459:
8439:
8415:
8390:
8375:
8303:
8244:
8220:
8195:
8170:
8159:from the original on 2013-11-05
8057:
7865:from the original on 2012-07-28
7789:from the original on 5 May 2012
7541:
7466:and the circumference formula 2
7370:(so the region it encloses has
6174:
6102:
6044:
5974:
3920:
1555:, the formula for the area is:
1256:Areas determined using calculus
1218:Karl Georg Christian von Staudt
1015:Area of a circle § History
8926:10.1080/00029890.2008.11920581
8826:"Filling Riemannian manifolds"
8788:The fractal geometry of nature
8785:Mandelbrot, Benoît B. (1983).
7986:
7918:
7888:; Otfried Schwarzkopf (2000).
7739:Area of a convex quadrilateral
7329:(rotation around the x-axis)
7293:
7287:
7262:
7256:
7127:
7091:
7040:
7028:
6974:
6947:
6813:
6807:
6787:
6781:
6429:
6414:
6399:
6384:
6337:
6322:
6229:
6209:
6154:
6139:
6082:
6067:
6024:
6009:
5843:
5827:
5703:
5691:
5394:
5376:
5328:
5316:
5313:
5301:
5298:
5286:
5230:
5224:
4868:
4856:
4811:
4799:
4582:
4570:
4547:
4535:
4469:
4442:
4246:
4234:
4101:{\displaystyle \pi r(r+l)\,\!}
4093:
4081:
3908:
3895:
3847:
3821:
3772:
3760:
3757:
3745:
3705:
3699:
3690:
3684:
3618:
3601:
3519:
3483:
3303:
3290:
3284:
3272:
3259:
3253:
3227:
3224:
3218:
3209:
3203:
3197:
3191:
3185:
3179:
3082:
3076:
3060:An area bounded by a function
3042:
3036:
3003:
3000:
2994:
2985:
2979:
2973:
2878:
2872:
2660:
2522:
2472:
2466:
2422:
2404:
2352:
2340:
2337:
2325:
2322:
2310:
2182:Areas of 2-dimensional figures
2101:of the same height and radius.
1799:The formula for the area of a
1481:
1423:
1331:
1305:
1008:
742:
635:. If there is a unique number
584:. If the rectangle has length
13:
1:
9012:Angular/rotational quantities
8913:American Mathematical Monthly
8595:"The Surveyor's Area Formula"
8446:Clark, Walter Eugene (1930).
8282:A Manual of Greek Mathematics
8202:Mishra, Sunita (2023-06-13).
7750:
7697:List of triangle inequalities
7550:. Familiar examples include
7358:Relation of area to perimeter
7046:{\displaystyle A=2\pi r(r+h)}
6980:{\displaystyle A=2(ab+ac+bc)}
1781:A circle can be divided into
1526:
1337:{\displaystyle (x_{i},y_{i})}
1291:For a non-self-intersecting (
996:1 Decimal = 48.4 square yards
758:1 metre × 1 metre = 1 m
257:There are several well-known
229:International System of Units
8364:{\displaystyle A=\pi r^{2}.}
6728:{\displaystyle A=\pi ab\,\!}
5653:{\displaystyle A=ah_{a}\,\!}
4842:{\displaystyle \mathbf {r} }
4263:is the radius of a base and
3088:{\displaystyle r=r(\theta )}
2800:), between two points (here
1823:, and the width is half the
1760:as well as more complicated
1108:Area of a triangle § History
292:For a solid shape such as a
7:
8920:(8): 679–689: Theorem 4.1.
8896:, Dover Publ., pp. 379–380.
7673:
7484:
4426:is the length of the slant.
4370:is the height of the prism.
4252:{\displaystyle 2\pi r(r+h)}
1995:
1033:constant of proportionality
1031:, but did not identify the
10:
9999:
9585:{\displaystyle {\dot {m}}}
9009:
8492:Bourke, Paul (July 1988).
7505:
2143:On the Sphere and Cylinder
2075:
1999:
1792:
1772:
1634:
1284:
1156:from the classical age of
1105:
1012:
1003:
999:1 Chatak = 180 square feet
965:1 barn = 10 square meters.
892:
658:for all such step regions
345:
29:
9283:specific angular momentum
9011:
9006:
9001:
8892:Dorrie, Heinrich (1965),
8666:10.1017/S0025557200182270
8565:. CRC Press. p. 43.
8382:Heath, Thomas L. (1921).
8044:Real and Complex Analysis
7724:—A list of areas by size.
7702:One-seventh area triangle
7680:Brahmagupta quadrilateral
4556:{\displaystyle z=f(x,y),}
4418:is the area of the base,
4293:can also be rewritten as
4212:is the length of an edge.
4139:{\displaystyle \pi r^{2}}
2061:{\displaystyle A=\pi xy.}
1876:. In ancient times, the
1665:can be subdivided into a
1611:. On the other hand, if
1586:is given by the formula:
1067:. (The circumference is 2
911:1 are = 100 square metres
180:refers to the area of an
106:
90:
78:
62:
46:
41:
8824:Gromov, Mikhael (1983).
8117:. Addison-Wesley Pub. Co
8026:. Prentice-Hall. p. 98,
7896:(2nd revised ed.).
7580:isoperimetric inequality
7364:isoperimetric inequality
2105:Most basic formulas for
1214:Carl Anton Bretschneider
1042:Subsequently, Book I of
1019:In the 5th century BCE,
687:
227:of a fixed size. In the
190:three-dimensional object
166:refers to the area of a
9407:rotational acceleration
8310:Stewart, James (2003).
8063:Gerald Folland (1999).
7003:(incl. bottom and top)
5067:{\displaystyle A=s^{2}}
4362:is the area of a base,
4146:is the base area while
3163:The area enclosed by a
2491:is the given angle and
2072:Non-planar surface area
1222:Bretschneider's formula
1193:In the 7th century CE,
1117:found what is known as
1064:Measurement of a Circle
9586:
8852:10.4310/jdg/1214509283
8528:; Chein, Orin (2012).
8365:
7894:Computational Geometry
7664:
7622:
7420:
7354:" (shoelace formula).
7321:
7197:
7134:
7047:
6981:
6910:
6826:
6729:
6680:
6594:
6560:
6486:
6459:
6436:
6347:
6280:
6279:{\displaystyle p=na\ }
6252:
6238:
6161:
6089:
6031:
5960:
5898:
5862:
5789:
5721:
5654:
5601:
5543:
5472:
5401:
5339:
5239:
5168:
5112:
5068:
5006:
4896:
4843:
4821:
4766:
4630:
4610:
4557:
4496:
4476:
4408:
4352:
4310:
4287:
4286:{\displaystyle 2\pi r}
4253:
4202:
4201:{\displaystyle 6s^{2}}
4166:
4165:{\displaystyle \pi rl}
4140:
4102:
4062:
4012:
3934:
3779:
3643:
3533:
3310:
3234:
3153:
3089:
3049:
3020:
2895:
2817:
2809:
2761:
2667:
2479:
2429:
2361:
2270:
2233:
2227:
2102:
2062:
1986:
1790:
1746:
1695:
1686: (parallelogram).
1650:
1540:
1495:
1422:
1338:
1095:circumscribed polygons
895:Category:Units of area
782:1 square centimetre =
752:
701:
306:multivariable calculus
136:
9587:
8559:Joshi, K. D. (2002).
8366:
8152:. 8th ed. Chapter 5.
8109:Moise, Edwin (1963).
8084:Apostol, Tom (1967).
8042:Walter Rudin (1966).
7665:
7623:
7421:
7322:
7215:Surface of revolution
7198:
7135:
7048:
6982:
6911:
6827:
6730:
6681:
6595:
6593:{\displaystyle d=2r:}
6561:
6487:
6460:
6437:
6348:
6281:
6251:
6239:
6162:
6090:
6032:
5961:
5899:
5863:
5790:
5722:
5655:
5602:
5544:
5473:
5402:
5340:
5240:
5169:
5113:
5069:
5007:
4897:
4844:
4822:
4767:
4631:
4611:
4558:
4497:
4495:{\displaystyle \ell }
4477:
4409:
4353:
4351:{\displaystyle 2B+Ph}
4311:
4309:{\displaystyle \pi d}
4288:
4254:
4203:
4167:
4141:
4103:
4063:
4013:
3935:
3805:, we can obtain that
3780:
3644:
3534:
3311:
3235:
3154:
3090:
3050:
3021:
2905:The area between the
2896:
2815:
2791:
2762:
2668:
2480:
2430:
2362:
2271:
2228:
2189:
2085:
2063:
1987:
1780:
1768:Area of curved shapes
1747:
1693:
1648:
1534:
1496:
1396:
1339:
1297:Cartesian coordinates
1252:in the 19th century.
1234:Cartesian coordinates
1199:Brahmagupta's formula
768:1 square kilometre =
750:
695:
325:differential geometry
137:
32:Area (disambiguation)
9567:
9417:angular acceleration
9189:angular displacement
8653:Mathematical Gazette
8336:
7635:
7596:
7557:The question of the
7388:
7222:
7159:
7076:
7010:
6935:
6865:
6754:
6705:
6624:
6572:
6509:
6473:
6446:
6357:
6295:
6258:
6171:
6099:
6041:
5971:
5911:
5888:
5815:
5747:
5679:
5626:
5572:
5506:
5497:equilateral triangle
5419:
5355:
5272:
5193:
5137:
5111:{\displaystyle A=ab}
5093:
5045:
4909:
4853:
4831:
4783:
4643:
4620:
4567:
4523:
4486:
4436:
4380:
4330:
4297:
4271:
4222:
4182:
4150:
4120:
4072:
4030:
3956:
3809:
3678:
3553:
3330:
3244:
3170:
3107:
3064:
3048:{\displaystyle f(x)}
3030:
2949:
2909:of two functions is
2845:
2732:
2507:
2439:
2383:
2302:
2245:
2194:
2111:developable surfaces
2037:
1895:
1878:method of exhaustion
1717:
1615:is developed before
1523:and so refers to 0.
1370:
1302:
1228:General polygon area
1203:cyclic quadrilateral
1201:, for the area of a
1139:Elements of Geometry
1021:Hippocrates of Chios
115:
9507:weighted position:
9324:rotational velocity
9272:kinematic viscosity
8992:Classical mechanics
8251:www.clicbrics.com.
7722:Orders of magnitude
7251:
6777:
5022:
4779:in the vector form
4267:is the height. The
3672:quadratic functions
3594:
3482:
3418:
3361:
2972:
2868:
2146:. The formula is:
2002:Ellipse § Area
1926:
1516:+1 is expressed as
1287:Polygon § Area
1260:The development of
1232:The development of
1115:Heron of Alexandria
1087:his doubling method
1029:lune of Hippocrates
786:square millimetres.
210:squaring the circle
9582:
8881:Mathematical Plums
8753:Weisstein, Eric W.
8399:Weisstein, Eric W.
8361:
8020:do Carmo, Manfredo
7961:Weisstein, Eric W.
7776:Weisstein, Eric W.
7660:
7618:
7520:at the triangle's
7416:
7352:Surveyor's formula
7317:
7237:
7193:
7130:
7043:
6977:
6906:
6822:
6763:
6725:
6676:
6590:
6556:
6485:{\displaystyle R:}
6482:
6458:{\displaystyle r:}
6455:
6432:
6427:
6397:
6370:
6343:
6335:
6314:
6276:
6253:
6234:
6227:
6188:
6157:
6152:
6121:
6085:
6080:
6027:
6022:
5988:
5956:
5894:
5858:
5785:
5717:
5650:
5597:
5539:
5468:
5412:Isosceles triangle
5397:
5374:
5335:
5235:
5164:
5108:
5064:
5020:
5002:
4892:
4839:
4817:
4777:parametric surface
4762:
4626:
4606:
4553:
4506:is the width, and
4492:
4472:
4404:
4348:
4306:
4283:
4249:
4198:
4162:
4136:
4112:is the radius and
4098:
4058:
4008:
3930:
3775:
3661:mechanical device.
3653:(For details, see
3639:
3566:
3529:
3454:
3390:
3333:
3306:
3230:
3149:
3085:
3045:
3016:
2958:
2891:
2854:
2818:
2810:
2757:
2663:
2520:
2475:
2452:
2425:
2402:
2357:
2266:
2258:
2234:
2223:
2221:
2134:Gaussian curvature
2103:
2058:
1982:
1909:
1827:of the circle, or
1791:
1742:
1696:
1651:
1568: (rectangle).
1541:
1491:
1361:surveyor's formula
1334:
1242:surveyor's formula
1189:Quadrilateral area
1158:Indian mathematics
1055:Euclidean geometry
1053:used the tools of
990:1 Ghumaon = 1 acre
753:
702:
132:
94:SI base units
9970:
9969:
9965:
9964:
9579:
9532:moment of inertia
8798:978-0-7167-1186-5
8760:Wolfram MathWorld
8729:978-602-00-0090-9
8699:978-979-758-477-1
8572:978-0-8493-1319-6
8545:978-0-486-13174-0
8534:. Dover. p.
8402:"Heron's Formula"
8327:978-0-534-39330-4
8292:978-0-486-43231-1
8257:www.clicbrics.com
8232:timesproperty.com
8032:978-0-13-212589-5
7968:Wolfram MathWorld
7941:978-0-486-60509-8
7911:978-3-540-65620-3
7783:Wolfram MathWorld
7691:Heronian triangle
7655:
7652:
7616:
7613:
7578:A version of the
7563:Riemannian circle
7496:fractal dimension
7341:
7340:
7307:
7125:
6803:
6672:
6641:
6554:
6426:
6396:
6369:
6334:
6313:
6275:
6226:
6187:
6151:
6120:
6079:
6021:
5987:
5954:
5936:
5897:{\displaystyle n}
5841:
5771:
5764:
5713:
5589:
5525:
5521:
5466:
5436:
5373:
5331:
5210:
5154:
4978:
4953:
4743:
4725:
4687:
4629:{\displaystyle D}
4430:Rectangular prism
4402:
4001:
3893:
3880:
3626:
3621:
3604:
3564:
3516:
3498:
3452:
3431:
3374:
3287:
3256:
3182:
3124:
3097:polar coordinates
2749:
2717:to find the area.
2519:
2451:
2401:
2355:
2280:is any side, and
2257:
2220:
1952:
1886:definite integral
1882:integral calculus
1752: (triangle).
1734:
1387:
1262:integral calculus
1037:Eudoxus of Cnidus
884:square kilometres
775:1 square metre =
714:square kilometres
342:Formal definition
145:
144:
16:(Redirected from
9990:
9957:
9953:
9941:
9937:
9917:
9913:
9901:
9897:
9875:
9871:
9861:
9851:
9836:
9832:
9822:
9800:
9796:
9786:
9776:
9761:
9757:
9744:
9720:
9714:
9706:
9695:
9689:
9678:
9671:angular momentum
9656:
9650:
9642:
9627:
9621:
9611:
9598:
9592:
9591:
9589:
9588:
9583:
9581:
9580:
9572:
9545:
9539:
9523:
9500:
9481:
9476:
9454:
9432:
9426:
9409:
9393:
9371:
9365:
9356:angular velocity
9353:
9334:
9333:
9320:
9319:
9312:rotational speed
9308:
9307:
9290:
9279:
9264:
9254:
9235:
9234:
9217:
9212:
9198:
9186:
9166:
9147:
9146:
9140:
9134:
9123:
9097:
9077:
9062:
9046:
9039:
9004:
9003:
8985:
8978:
8971:
8962:
8961:
8950:
8949:
8903:
8897:
8890:
8884:
8877:
8868:
8867:
8845:
8821:
8815:
8814:
8812:
8810:
8782:
8776:
8775:
8773:
8771:
8749:
8738:
8737:
8714:
8708:
8707:
8684:
8678:
8677:
8660:(522): 536–540.
8647:
8641:
8640:
8638:
8636:
8630:
8599:
8590:
8581:
8580:
8556:
8550:
8549:
8526:Averbach, Bonnie
8522:
8516:
8515:
8513:
8511:
8505:
8498:
8489:
8483:
8482:
8472:
8463:
8457:
8456:
8454:
8443:
8437:
8436:
8434:
8433:
8419:
8413:
8412:
8411:
8394:
8388:
8387:
8379:
8373:
8372:
8370:
8368:
8367:
8362:
8357:
8356:
8307:
8301:
8300:
8277:Heath, Thomas L.
8273:
8267:
8266:
8264:
8263:
8248:
8242:
8241:
8239:
8238:
8224:
8218:
8217:
8215:
8214:
8199:
8193:
8192:
8190:
8189:
8182:Magicbricks Blog
8174:
8168:
8167:
8165:
8164:
8158:
8151:
8140:
8127:
8126:
8124:
8122:
8116:
8106:
8100:
8099:
8081:
8075:
8061:
8055:
8040:
8034:
8017:
8008:
8007:
8005:
8004:
7998:CK-12 Foundation
7990:
7984:
7983:
7981:
7979:
7957:
7946:
7945:
7922:
7916:
7915:
7881:
7875:
7874:
7872:
7870:
7851:
7842:
7841:
7839:
7837:
7822:
7799:
7798:
7796:
7794:
7772:
7744:Robbins pentagon
7669:
7667:
7666:
7661:
7656:
7654:
7653:
7648:
7639:
7627:
7625:
7624:
7619:
7617:
7615:
7614:
7609:
7600:
7498:of the fractal.
7425:
7423:
7422:
7417:
7412:
7411:
7337:
7326:
7324:
7323:
7318:
7313:
7308:
7306:
7305:
7300:
7296:
7286:
7266:
7250:
7245:
7209:
7202:
7200:
7199:
7194:
7180:
7179:
7146:
7139:
7137:
7136:
7131:
7126:
7124:
7123:
7111:
7110:
7101:
7059:
7052:
7050:
7049:
7044:
6993:
6986:
6984:
6983:
6978:
6922:
6915:
6913:
6912:
6907:
6905:
6904:
6889:
6888:
6838:
6831:
6829:
6828:
6823:
6801:
6794:
6776:
6771:
6741:
6734:
6732:
6731:
6726:
6692:
6685:
6683:
6682:
6677:
6673:
6668:
6657:
6652:
6651:
6642:
6634:
6611:
6599:
6597:
6596:
6591:
6565:
6563:
6562:
6557:
6555:
6550:
6549:
6548:
6535:
6530:
6529:
6491:
6489:
6488:
6483:
6464:
6462:
6461:
6456:
6441:
6439:
6438:
6433:
6428:
6419:
6398:
6389:
6371:
6362:
6352:
6350:
6349:
6344:
6336:
6327:
6315:
6306:
6285:
6283:
6282:
6277:
6273:
6243:
6241:
6240:
6235:
6228:
6222:
6214:
6202:
6201:
6189:
6180:
6166:
6164:
6163:
6158:
6153:
6144:
6132:
6131:
6122:
6119:
6108:
6094:
6092:
6091:
6086:
6081:
6072:
6060:
6059:
6036:
6034:
6033:
6028:
6023:
6014:
6002:
6001:
5989:
5980:
5965:
5963:
5962:
5957:
5955:
5950:
5942:
5937:
5932:
5924:
5903:
5901:
5900:
5895:
5874:
5867:
5865:
5864:
5859:
5855:
5854:
5842:
5837:
5801:
5794:
5792:
5791:
5786:
5782:
5781:
5772:
5767:
5765:
5757:
5733:
5726:
5724:
5723:
5718:
5714:
5709:
5689:
5666:
5659:
5657:
5656:
5651:
5647:
5646:
5613:
5606:
5604:
5603:
5598:
5590:
5582:
5555:
5548:
5546:
5545:
5540:
5536:
5535:
5526:
5517:
5516:
5484:
5477:
5475:
5474:
5469:
5467:
5465:
5464:
5452:
5451:
5439:
5437:
5429:
5406:
5404:
5403:
5398:
5375:
5366:
5351:
5344:
5342:
5341:
5336:
5332:
5282:
5251:
5244:
5242:
5241:
5236:
5211:
5203:
5180:
5173:
5171:
5170:
5165:
5155:
5147:
5124:
5117:
5115:
5114:
5109:
5080:
5073:
5071:
5070:
5065:
5063:
5062:
5023:
5019:
5016:List of formulas
5011:
5009:
5008:
5003:
4984:
4980:
4979:
4977:
4969:
4968:
4959:
4954:
4952:
4944:
4943:
4934:
4927:
4926:
4901:
4899:
4898:
4893:
4891:
4890:
4885:
4848:
4846:
4845:
4840:
4838:
4826:
4824:
4823:
4818:
4798:
4790:
4771:
4769:
4768:
4763:
4744:
4736:
4735:
4730:
4726:
4724:
4716:
4708:
4698:
4697:
4692:
4688:
4686:
4678:
4670:
4663:
4661:
4660:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4605:
4604:
4599:
4562:
4560:
4559:
4554:
4501:
4499:
4498:
4493:
4481:
4479:
4478:
4473:
4413:
4411:
4410:
4405:
4403:
4398:
4390:
4357:
4355:
4354:
4349:
4320:is the diameter.
4315:
4313:
4312:
4307:
4292:
4290:
4289:
4284:
4258:
4256:
4255:
4250:
4207:
4205:
4204:
4199:
4197:
4196:
4171:
4169:
4168:
4163:
4145:
4143:
4142:
4137:
4135:
4134:
4107:
4105:
4104:
4099:
4067:
4065:
4064:
4059:
4045:
4044:
4017:
4015:
4014:
4009:
4007:
4003:
4002:
4000:
3999:
3987:
3986:
3977:
3939:
3937:
3936:
3931:
3916:
3915:
3894:
3886:
3881:
3879:
3878:
3877:
3864:
3863:
3862:
3858:
3833:
3832:
3819:
3784:
3782:
3781:
3776:
3723:
3722:
3648:
3646:
3645:
3640:
3628:
3627:
3622:
3614:
3612:
3606:
3605:
3597:
3593:
3592:
3591:
3581:
3580:
3579:
3565:
3557:
3538:
3536:
3535:
3530:
3518:
3517:
3509:
3500:
3499:
3491:
3481:
3480:
3479:
3469:
3468:
3467:
3453:
3445:
3433:
3432:
3424:
3417:
3416:
3415:
3405:
3404:
3403:
3376:
3375:
3367:
3360:
3359:
3358:
3348:
3347:
3346:
3316:is given by the
3315:
3313:
3312:
3307:
3302:
3301:
3289:
3288:
3280:
3271:
3270:
3258:
3257:
3249:
3239:
3237:
3236:
3231:
3184:
3183:
3175:
3165:parametric curve
3158:
3156:
3155:
3150:
3138:
3137:
3125:
3117:
3094:
3092:
3091:
3086:
3054:
3052:
3051:
3046:
3025:
3023:
3022:
3017:
2971:
2966:
2900:
2898:
2897:
2892:
2867:
2862:
2784:Area in calculus
2766:
2764:
2763:
2758:
2750:
2742:
2675:shoelace formula
2672:
2670:
2669:
2664:
2659:
2658:
2649:
2648:
2636:
2635:
2626:
2625:
2613:
2612:
2603:
2602:
2590:
2589:
2580:
2579:
2567:
2566:
2557:
2556:
2544:
2543:
2534:
2533:
2521:
2512:
2502:
2496:
2490:
2484:
2482:
2481:
2476:
2453:
2444:
2434:
2432:
2431:
2426:
2403:
2394:
2366:
2364:
2363:
2358:
2356:
2306:
2275:
2273:
2272:
2267:
2259:
2250:
2232:
2230:
2229:
2224:
2222:
2216:
2205:
2177:General formulas
2168:
2158:
2067:
2065:
2064:
2059:
2030:the formula is:
2029:
2023:
1991:
1989:
1988:
1983:
1978:
1977:
1953:
1951:
1950:
1938:
1937:
1928:
1925:
1920:
1868:
1853:
1840:
1833:
1822:
1812:
1795:Area of a circle
1751:
1749:
1748:
1743:
1735:
1727:
1685:
1598:
1585:
1581:
1567:
1554:
1550:
1538:
1500:
1498:
1497:
1492:
1490:
1489:
1480:
1479:
1470:
1469:
1451:
1450:
1435:
1434:
1421:
1410:
1395:
1394:
1388:
1380:
1343:
1341:
1340:
1335:
1330:
1329:
1317:
1316:
1281:Polygon formulas
1162:Indian astronomy
1084:
1077:
1070:
987:1 Killa = 1 acre
883:
882:
879:
876:
871:1 square mile =
867:
866:
863:
858:1 square yard =
854:
853:
850:
845:1 square foot =
827:1 square yard =
791:Non-metric units
700:made of PVC pipe
679:
657:
634:
605:
576:Every rectangle
572:
542:is congruent to
526:
491:
481:
456:
413:
403:
378:
333:Lebesgue measure
277:can be found by
141:
139:
138:
133:
131:
130:
125:
124:
95:
51:
39:
38:
21:
9998:
9997:
9993:
9992:
9991:
9989:
9988:
9987:
9973:
9972:
9971:
9966:
9955:
9954:
9949:
9939:
9938:
9933:
9915:
9914:
9909:
9899:
9898:
9893:
9873:
9872:
9867:
9857:
9847:
9834:
9833:
9828:
9816:
9798:
9797:
9792:
9782:
9772:
9759:
9758:
9756:
9750:
9740:
9718:
9715:
9712:
9704:
9693:
9690:
9684:
9681:angular impulse
9674:
9654:
9651:
9648:
9640:
9625:
9622:
9617:
9607:
9596:
9593:
9571:
9570:
9568:
9565:
9564:
9563:
9543:
9540:
9535:
9508:
9501:
9496:
9479:
9477:
9470:
9455:
9450:
9430:
9427:
9420:
9410:
9405:
9394:
9389:
9369:
9366:
9359:
9349:
9335:
9327:
9322:
9315:
9310:
9303:
9298:
9291:
9286:
9281:
9275:
9265:
9260:
9250:
9236:
9230:
9225:
9213:
9210:
9205:
9199:
9192:
9182:
9167:
9162:
9152:
9142:
9136:
9130:
9125:
9119:
9098:
9093:
9078:
9073:
9063:
9058:
9042:
9035:
8997:
8989:
8959:
8954:
8953:
8904:
8900:
8891:
8887:
8878:
8871:
8843:10.1.1.400.9154
8822:
8818:
8808:
8806:
8799:
8783:
8779:
8769:
8767:
8750:
8741:
8730:
8716:
8715:
8711:
8700:
8686:
8685:
8681:
8648:
8644:
8634:
8632:
8628:
8614:10.2307/2686282
8597:
8591:
8584:
8573:
8557:
8553:
8546:
8523:
8519:
8509:
8507:
8503:
8496:
8490:
8486:
8470:
8464:
8460:
8452:
8444:
8440:
8431:
8429:
8427:Central College
8421:
8420:
8416:
8395:
8391:
8380:
8376:
8352:
8348:
8337:
8334:
8333:
8328:
8308:
8304:
8293:
8274:
8270:
8261:
8259:
8249:
8245:
8236:
8234:
8226:
8225:
8221:
8212:
8210:
8200:
8196:
8187:
8185:
8176:
8175:
8171:
8162:
8160:
8156:
8149:
8141:
8130:
8120:
8118:
8107:
8103:
8096:
8082:
8078:
8062:
8058:
8046:, McGraw-Hill,
8041:
8037:
8018:
8011:
8002:
8000:
7992:
7991:
7987:
7977:
7975:
7958:
7949:
7942:
7923:
7919:
7912:
7898:Springer-Verlag
7882:
7878:
7868:
7866:
7853:
7852:
7845:
7835:
7833:
7826:"Area Formulas"
7824:
7823:
7802:
7792:
7790:
7773:
7758:
7753:
7712:Routh's theorem
7676:
7647:
7643:
7638:
7636:
7633:
7632:
7608:
7604:
7599:
7597:
7594:
7593:
7548:minimal surface
7544:
7510:
7504:
7487:
7475:regular polygon
7407:
7403:
7389:
7386:
7385:
7377:) and for area
7360:
7327:
7309:
7301:
7279:
7278:
7274:
7273:
7265:
7246:
7241:
7223:
7220:
7219:
7175:
7171:
7160:
7157:
7156:
7119:
7115:
7106:
7102:
7100:
7077:
7074:
7073:
7069:(incl. bottom)
7067:
7011:
7008:
7007:
7001:
6936:
6933:
6932:
6900:
6896:
6884:
6880:
6866:
6863:
6862:
6859:
6790:
6772:
6767:
6755:
6752:
6751:
6706:
6703:
6702:
6658:
6656:
6647:
6643:
6633:
6625:
6622:
6621:
6617:Circular sector
6573:
6570:
6569:
6566:
6544:
6540:
6536:
6534:
6525:
6521:
6510:
6507:
6506:
6474:
6471:
6470:
6469:
6447:
6444:
6443:
6442:
6417:
6387:
6360:
6358:
6355:
6354:
6353:
6325:
6304:
6296:
6293:
6292:
6291:
6259:
6256:
6255:
6215:
6212:
6197:
6193:
6178:
6172:
6169:
6168:
6167:
6142:
6127:
6123:
6112:
6106:
6100:
6097:
6096:
6095:
6070:
6055:
6051:
6042:
6039:
6038:
6037:
6012:
5997:
5993:
5978:
5972:
5969:
5968:
5966:
5943:
5941:
5925:
5923:
5912:
5909:
5908:
5889:
5886:
5885:
5882:
5880:Regular polygon
5850:
5846:
5836:
5816:
5813:
5812:
5777:
5773:
5766:
5756:
5748:
5745:
5744:
5690:
5688:
5680:
5677:
5676:
5642:
5638:
5627:
5624:
5623:
5581:
5573:
5570:
5569:
5531:
5527:
5515:
5507:
5504:
5503:
5493:
5460:
5456:
5447:
5443:
5438:
5428:
5420:
5417:
5416:
5364:
5356:
5353:
5352:
5281:
5273:
5270:
5269:
5263:Heron's formula
5259:
5202:
5194:
5191:
5190:
5146:
5138:
5135:
5134:
5094:
5091:
5090:
5058:
5054:
5046:
5043:
5042:
5018:
4970:
4964:
4960:
4958:
4945:
4939:
4935:
4933:
4932:
4928:
4922:
4918:
4910:
4907:
4906:
4886:
4881:
4880:
4854:
4851:
4850:
4834:
4832:
4829:
4828:
4794:
4786:
4784:
4781:
4780:
4731:
4717:
4709:
4707:
4703:
4702:
4693:
4679:
4671:
4669:
4665:
4664:
4662:
4656:
4652:
4644:
4641:
4640:
4621:
4618:
4617:
4600:
4595:
4594:
4568:
4565:
4564:
4524:
4521:
4520:
4517:
4502:is the length,
4487:
4484:
4483:
4437:
4434:
4433:
4391:
4389:
4381:
4378:
4377:
4331:
4328:
4327:
4298:
4295:
4294:
4272:
4269:
4268:
4223:
4220:
4219:
4192:
4188:
4183:
4180:
4179:
4151:
4148:
4147:
4130:
4126:
4121:
4118:
4117:
4073:
4070:
4069:
4040:
4036:
4031:
4028:
4027:
3995:
3991:
3982:
3978:
3976:
3969:
3965:
3957:
3954:
3953:
3946:
3911:
3907:
3885:
3873:
3869:
3865:
3854:
3850:
3846:
3828:
3824:
3820:
3818:
3810:
3807:
3806:
3803:Vieta's formula
3718:
3714:
3679:
3676:
3675:
3668:
3613:
3611:
3610:
3596:
3595:
3587:
3583:
3582:
3575:
3571:
3570:
3556:
3554:
3551:
3550:
3508:
3507:
3490:
3489:
3475:
3471:
3470:
3463:
3459:
3458:
3444:
3423:
3422:
3411:
3407:
3406:
3399:
3395:
3394:
3366:
3365:
3354:
3350:
3349:
3342:
3338:
3337:
3331:
3328:
3327:
3297:
3293:
3279:
3278:
3266:
3262:
3248:
3247:
3245:
3242:
3241:
3240:with endpoints
3174:
3173:
3171:
3168:
3167:
3133:
3129:
3116:
3108:
3105:
3104:
3065:
3062:
3061:
3031:
3028:
3027:
2967:
2962:
2950:
2947:
2946:
2863:
2858:
2846:
2843:
2842:
2786:
2741:
2733:
2730:
2729:
2710:
2706:
2698:
2694:
2686:
2682:
2654:
2650:
2644:
2640:
2631:
2627:
2621:
2617:
2608:
2604:
2598:
2594:
2585:
2581:
2575:
2571:
2562:
2558:
2552:
2548:
2539:
2535:
2529:
2525:
2510:
2508:
2505:
2504:
2498:
2492:
2486:
2442:
2440:
2437:
2436:
2392:
2384:
2381:
2380:
2305:
2303:
2300:
2299:
2295:Heron's formula
2248:
2246:
2243:
2242:
2206:
2203:
2195:
2192:
2191:
2184:
2179:
2164:
2159: (sphere),
2150:
2080:
2074:
2038:
2035:
2034:
2025:
2019:
2004:
1998:
1973:
1969:
1946:
1942:
1933:
1929:
1927:
1921:
1913:
1896:
1893:
1892:
1863:
1854: (circle).
1845:
1835:
1828:
1818:
1808:
1797:
1775:
1770:
1726:
1718:
1715:
1714:
1677:
1643:
1635:Main articles:
1633:
1599: (square).
1590:
1583:
1573:
1559:
1552:
1548:
1536:
1529:
1485:
1484:
1475:
1471:
1459:
1455:
1440:
1436:
1430:
1426:
1411:
1400:
1390:
1389:
1379:
1371:
1368:
1367:
1325:
1321:
1312:
1308:
1303:
1300:
1299:
1295:) polygon, the
1289:
1283:
1278:
1258:
1230:
1191:
1186:
1185:
1180:"), written by
1174:Shushu Jiuzhang
1119:Heron's formula
1111:
1103:
1082:
1075:
1068:
1017:
1011:
1006:
971:nuclear physics
897:
891:
880:
877:
874:
872:
864:
861:
859:
851:
848:
846:
793:
745:
737:SI derived unit
696:A square metre
690:
667:
640:
622:
593:
555:
497:
483:
473:
415:
405:
395:
369:
350:
344:
126:
120:
119:
118:
116:
113:
112:
93:
65:
58:
35:
28:
23:
22:
18:Area (geometry)
15:
12:
11:
5:
9996:
9986:
9985:
9968:
9967:
9963:
9962:
9943:
9927:
9925:
9922:
9903:
9887:
9885:
9881:
9880:
9841:
9810:
9808:
9805:
9766:
9754:
9734:
9732:
9728:
9727:
9698:
9668:
9666:
9663:
9634:
9601:
9578:
9575:
9560:Mass flow rate
9557:
9553:
9552:
9550:
9548:
9529:
9526:
9524:
9505:
9490:
9486:
9485:
9483:
9464:
9462:
9459:
9457:
9444:
9442:
9438:
9437:
9435:
9414:
9403:
9400:
9398:
9383:
9381:
9377:
9376:
9374:
9343:
9296:
9293:
9269:
9244:
9223:
9219:
9218:
9203:
9176:
9174:
9171:
9156:
9113:
9111:
9107:
9106:
9104:
9102:
9087:
9084:
9082:
9067:
9052:
9048:
9047:
9040:
9033:
9030:
9027:
9024:
9021:
9018:
9014:
9013:
9010:
9008:
9002:
8999:
8998:
8988:
8987:
8980:
8973:
8965:
8958:
8957:External links
8955:
8952:
8951:
8898:
8885:
8869:
8816:
8797:
8777:
8739:
8728:
8709:
8698:
8679:
8642:
8608:(4): 326–337.
8582:
8571:
8551:
8544:
8517:
8484:
8458:
8438:
8414:
8389:
8374:
8360:
8355:
8351:
8347:
8344:
8341:
8326:
8302:
8291:
8268:
8243:
8219:
8194:
8169:
8128:
8101:
8094:
8076:
8056:
8035:
8009:
7994:"Surface Area"
7985:
7964:"Surface Area"
7947:
7940:
7926:Boyer, Carl B.
7917:
7910:
7876:
7843:
7800:
7755:
7754:
7752:
7749:
7748:
7747:
7741:
7736:
7730:
7725:
7718:
7717:
7716:
7715:
7706:
7705:
7699:
7694:
7688:
7683:
7675:
7672:
7659:
7651:
7646:
7642:
7612:
7607:
7603:
7573:cyclic polygon
7565:remains open.
7543:
7540:
7506:Main article:
7503:
7502:Area bisectors
7500:
7486:
7483:
7473:The area of a
7427:
7426:
7415:
7410:
7406:
7402:
7399:
7396:
7393:
7359:
7356:
7339:
7338:
7331:
7316:
7312:
7304:
7299:
7295:
7292:
7289:
7285:
7282:
7277:
7272:
7269:
7264:
7261:
7258:
7255:
7249:
7244:
7240:
7236:
7233:
7230:
7227:
7217:
7211:
7210:
7203:
7192:
7189:
7186:
7183:
7178:
7174:
7170:
7167:
7164:
7154:
7148:
7147:
7140:
7129:
7122:
7118:
7114:
7109:
7105:
7099:
7096:
7093:
7090:
7087:
7084:
7081:
7071:
7061:
7060:
7053:
7042:
7039:
7036:
7033:
7030:
7027:
7024:
7021:
7018:
7015:
7005:
6995:
6994:
6987:
6976:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6940:
6930:
6924:
6923:
6916:
6903:
6899:
6895:
6892:
6887:
6883:
6879:
6876:
6873:
6870:
6860:
6853:
6852:
6850:
6843:
6840:
6839:
6832:
6821:
6818:
6815:
6812:
6809:
6806:
6800:
6797:
6793:
6789:
6786:
6783:
6780:
6775:
6770:
6766:
6762:
6759:
6749:
6743:
6742:
6735:
6722:
6719:
6716:
6713:
6710:
6700:
6694:
6693:
6686:
6671:
6667:
6664:
6661:
6655:
6650:
6646:
6640:
6637:
6632:
6629:
6619:
6613:
6612:
6605:
6589:
6586:
6583:
6580:
6577:
6553:
6547:
6543:
6539:
6533:
6528:
6524:
6520:
6517:
6514:
6504:
6498:
6497:
6481:
6478:
6454:
6451:
6431:
6425:
6422:
6416:
6413:
6410:
6407:
6404:
6401:
6395:
6392:
6386:
6383:
6380:
6377:
6374:
6368:
6365:
6342:
6339:
6333:
6330:
6324:
6321:
6318:
6312:
6309:
6303:
6300:
6272:
6269:
6266:
6263:
6245:
6231:
6225:
6221:
6218:
6211:
6208:
6205:
6200:
6196:
6192:
6186:
6183:
6177:
6156:
6150:
6147:
6141:
6138:
6135:
6130:
6126:
6118:
6115:
6111:
6105:
6084:
6078:
6075:
6069:
6066:
6063:
6058:
6054:
6050:
6047:
6026:
6020:
6017:
6011:
6008:
6005:
6000:
5996:
5992:
5986:
5983:
5977:
5953:
5949:
5946:
5940:
5935:
5931:
5928:
5922:
5919:
5916:
5906:
5893:
5876:
5875:
5868:
5853:
5849:
5845:
5840:
5835:
5832:
5829:
5826:
5823:
5820:
5810:
5803:
5802:
5795:
5780:
5776:
5770:
5763:
5760:
5755:
5752:
5742:
5735:
5734:
5727:
5712:
5708:
5705:
5702:
5699:
5696:
5693:
5687:
5684:
5674:
5668:
5667:
5660:
5645:
5641:
5637:
5634:
5631:
5621:
5615:
5614:
5607:
5596:
5593:
5588:
5585:
5580:
5577:
5567:
5557:
5556:
5549:
5534:
5530:
5524:
5520:
5514:
5511:
5501:
5486:
5485:
5478:
5463:
5459:
5455:
5450:
5446:
5442:
5435:
5432:
5427:
5424:
5414:
5408:
5407:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5372:
5369:
5363:
5360:
5345:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5280:
5277:
5267:
5253:
5252:
5245:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5209:
5206:
5201:
5198:
5188:
5182:
5181:
5174:
5161:
5158:
5153:
5150:
5145:
5142:
5132:
5126:
5125:
5118:
5107:
5104:
5101:
5098:
5088:
5082:
5081:
5074:
5061:
5057:
5053:
5050:
5040:
5034:
5033:
5030:
5027:
5017:
5014:
5013:
5012:
5001:
4998:
4995:
4991:
4988:
4983:
4976:
4973:
4967:
4963:
4957:
4951:
4948:
4942:
4938:
4931:
4925:
4921:
4917:
4914:
4889:
4884:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4837:
4816:
4813:
4810:
4807:
4804:
4801:
4797:
4793:
4789:
4773:
4772:
4761:
4758:
4755:
4751:
4748:
4742:
4739:
4734:
4729:
4723:
4720:
4715:
4712:
4706:
4701:
4696:
4691:
4685:
4682:
4677:
4674:
4668:
4659:
4655:
4651:
4648:
4625:
4603:
4598:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4516:
4513:
4512:
4511:
4510:is the height.
4491:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4427:
4401:
4397:
4394:
4388:
4385:
4371:
4347:
4344:
4341:
4338:
4335:
4321:
4305:
4302:
4282:
4279:
4276:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4213:
4195:
4191:
4187:
4173:
4161:
4158:
4155:
4133:
4129:
4125:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4057:
4054:
4051:
4048:
4043:
4039:
4035:
4006:
3998:
3994:
3990:
3985:
3981:
3975:
3972:
3968:
3964:
3961:
3945:
3942:
3929:
3926:
3923:
3919:
3914:
3910:
3906:
3903:
3900:
3897:
3892:
3889:
3884:
3876:
3872:
3868:
3861:
3857:
3853:
3849:
3845:
3842:
3839:
3836:
3831:
3827:
3823:
3817:
3814:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3721:
3717:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3667:
3664:
3663:
3662:
3651:
3650:
3649:
3638:
3635:
3632:
3625:
3620:
3617:
3609:
3603:
3600:
3590:
3586:
3578:
3574:
3569:
3563:
3560:
3547:-component of
3541:
3540:
3539:
3528:
3525:
3521:
3515:
3512:
3506:
3503:
3497:
3494:
3488:
3485:
3478:
3474:
3466:
3462:
3457:
3451:
3448:
3443:
3440:
3437:
3430:
3427:
3421:
3414:
3410:
3402:
3398:
3393:
3389:
3386:
3383:
3380:
3373:
3370:
3364:
3357:
3353:
3345:
3341:
3336:
3322:
3321:
3318:line integrals
3305:
3300:
3296:
3292:
3286:
3283:
3277:
3274:
3269:
3265:
3261:
3255:
3252:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3181:
3178:
3160:
3159:
3148:
3145:
3142:
3136:
3132:
3128:
3123:
3120:
3115:
3112:
3101:
3100:
3084:
3081:
3078:
3075:
3072:
3069:
3057:
3056:
3044:
3041:
3038:
3035:
3015:
3012:
3009:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2970:
2965:
2961:
2957:
2954:
2943:
2942:
2902:
2901:
2890:
2887:
2884:
2880:
2877:
2874:
2871:
2866:
2861:
2857:
2853:
2850:
2839:
2838:
2785:
2782:
2781:
2780:
2777:Pick's theorem
2756:
2753:
2748:
2745:
2740:
2737:
2722:simple polygon
2718:
2708:
2704:
2696:
2692:
2684:
2680:
2662:
2657:
2653:
2647:
2643:
2639:
2634:
2630:
2624:
2620:
2616:
2611:
2607:
2601:
2597:
2593:
2588:
2584:
2578:
2574:
2570:
2565:
2561:
2555:
2551:
2547:
2542:
2538:
2532:
2528:
2524:
2518:
2515:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2450:
2447:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2400:
2397:
2391:
2388:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2265:
2262:
2256:
2253:
2219:
2215:
2212:
2209:
2202:
2199:
2190:Triangle area
2183:
2180:
2178:
2175:
2161:
2160:
2076:Main article:
2073:
2070:
2069:
2068:
2057:
2054:
2051:
2048:
2045:
2042:
2000:Main article:
1997:
1994:
1993:
1992:
1981:
1976:
1972:
1968:
1964:
1960:
1957:
1949:
1945:
1941:
1936:
1932:
1924:
1919:
1916:
1912:
1908:
1904:
1900:
1856:
1855:
1793:Main article:
1774:
1771:
1769:
1766:
1754:
1753:
1741:
1738:
1733:
1730:
1725:
1722:
1688:
1687:
1671:right triangle
1632:
1629:
1621:multiplication
1601:
1600:
1570:
1569:
1528:
1525:
1502:
1501:
1488:
1483:
1478:
1474:
1468:
1465:
1462:
1458:
1454:
1449:
1446:
1443:
1439:
1433:
1429:
1425:
1420:
1417:
1414:
1409:
1406:
1403:
1399:
1393:
1386:
1383:
1378:
1375:
1333:
1328:
1324:
1320:
1315:
1311:
1307:
1285:Main article:
1282:
1279:
1277:
1274:
1257:
1254:
1238:René Descartes
1229:
1226:
1190:
1187:
1112:
1104:
1102:
1099:
1059:right triangle
1013:Main article:
1010:
1007:
1005:
1002:
1001:
1000:
997:
994:
991:
988:
967:
966:
952:
951:
925:
924:
913:
912:
890:
887:
886:
885:
869:
856:
843:
836:
835:
832:
821:
820:
813:
812:
792:
789:
788:
787:
780:
773:
744:
741:
706:unit of length
689:
686:
682:
681:
607:
574:
528:
458:
380:
348:Jordan measure
343:
340:
321:linear algebra
302:ancient Greeks
218:polygonal area
143:
142:
129:
123:
110:
104:
103:
97:
88:
87:
82:
76:
75:
66:
64:Common symbols
63:
60:
59:
52:
44:
43:
26:
9:
6:
4:
3:
2:
9995:
9984:
9981:
9980:
9978:
9961:
9952:
9947:
9944:
9936:
9931:
9928:
9926:
9923:
9921:
9912:
9907:
9904:
9896:
9891:
9888:
9886:
9883:
9882:
9879:
9870:
9865:
9860:
9855:
9850:
9845:
9842:
9840:
9831:
9826:
9821:
9820:
9814:
9811:
9809:
9806:
9804:
9795:
9790:
9785:
9780:
9775:
9770:
9767:
9765:
9753:
9748:
9743:
9738:
9735:
9733:
9730:
9729:
9726:
9722:
9710:
9702:
9699:
9697:
9688:
9682:
9677:
9672:
9669:
9667:
9664:
9662:
9658:
9646:
9638:
9635:
9633:
9629:
9620:
9615:
9610:
9605:
9602:
9600:
9576:
9573:
9561:
9558:
9555:
9554:
9551:
9549:
9547:
9538:
9533:
9530:
9527:
9525:
9522:
9519:
9515:
9511:
9506:
9504:
9499:
9494:
9491:
9488:
9487:
9484:
9475:
9474:
9468:
9465:
9463:
9460:
9458:
9453:
9448:
9445:
9443:
9440:
9439:
9436:
9434:
9425:
9424:
9418:
9415:
9413:
9408:
9404:
9401:
9399:
9397:
9392:
9387:
9384:
9382:
9379:
9378:
9375:
9373:
9364:
9363:
9357:
9352:
9347:
9346:angular speed
9344:
9342:
9338:
9332:
9331:
9325:
9318:
9313:
9306:
9301:
9297:
9294:
9289:
9284:
9278:
9273:
9270:
9268:
9263:
9258:
9253:
9248:
9245:
9243:
9239:
9233:
9228:
9224:
9221:
9220:
9216:
9208:
9204:
9202:
9197:
9196:
9190:
9185:
9180:
9177:
9175:
9172:
9170:
9165:
9160:
9157:
9155:
9151:
9145:
9139:
9133:
9128:
9122:
9117:
9114:
9112:
9109:
9108:
9105:
9103:
9101:
9096:
9091:
9088:
9085:
9083:
9081:
9076:
9071:
9068:
9066:
9061:
9056:
9053:
9050:
9049:
9045:
9041:
9038:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9015:
9005:
9000:
8996:
8993:
8986:
8981:
8979:
8974:
8972:
8967:
8966:
8963:
8947:
8943:
8939:
8935:
8931:
8927:
8923:
8919:
8915:
8914:
8909:
8902:
8895:
8889:
8882:
8876:
8874:
8865:
8861:
8857:
8853:
8849:
8844:
8839:
8835:
8831:
8827:
8820:
8804:
8800:
8794:
8791:. Macmillan.
8790:
8789:
8781:
8765:
8761:
8757:
8754:
8748:
8746:
8744:
8735:
8731:
8725:
8721:
8720:
8713:
8705:
8701:
8695:
8691:
8690:
8683:
8675:
8671:
8667:
8663:
8659:
8655:
8654:
8646:
8627:
8623:
8619:
8615:
8611:
8607:
8603:
8596:
8589:
8587:
8578:
8574:
8568:
8564:
8563:
8555:
8547:
8541:
8537:
8533:
8532:
8527:
8521:
8502:
8495:
8488:
8481:(5): 596–597.
8480:
8476:
8469:
8462:
8451:
8450:
8442:
8428:
8424:
8418:
8409:
8408:
8403:
8400:
8393:
8385:
8378:
8371:
8358:
8353:
8349:
8345:
8342:
8339:
8329:
8323:
8319:
8315:
8314:
8306:
8298:
8294:
8288:
8284:
8283:
8278:
8272:
8258:
8254:
8247:
8233:
8229:
8223:
8209:
8205:
8198:
8183:
8179:
8173:
8155:
8148:
8147:
8139:
8137:
8135:
8133:
8115:
8114:
8105:
8097:
8095:9780471000051
8091:
8087:
8080:
8074:
8073:0-471-31716-0
8070:
8066:
8060:
8053:
8052:0-07-100276-6
8049:
8045:
8039:
8033:
8029:
8025:
8021:
8016:
8014:
7999:
7995:
7989:
7973:
7969:
7965:
7962:
7956:
7954:
7952:
7943:
7937:
7933:
7932:
7927:
7921:
7913:
7907:
7903:
7899:
7895:
7891:
7887:
7886:Mark Overmars
7880:
7864:
7860:
7856:
7850:
7848:
7831:
7827:
7821:
7819:
7817:
7815:
7813:
7811:
7809:
7807:
7805:
7788:
7784:
7780:
7777:
7771:
7769:
7767:
7765:
7763:
7761:
7756:
7745:
7742:
7740:
7737:
7734:
7731:
7729:
7726:
7723:
7720:
7719:
7713:
7710:
7709:
7708:
7707:
7703:
7700:
7698:
7695:
7692:
7689:
7687:
7686:Equiareal map
7684:
7681:
7678:
7677:
7671:
7657:
7649:
7644:
7640:
7629:
7610:
7605:
7601:
7590:
7587:
7585:
7581:
7576:
7574:
7569:
7566:
7564:
7560:
7555:
7553:
7549:
7539:
7537:
7532:
7529:
7527:
7523:
7519:
7515:
7509:
7499:
7497:
7493:
7482:
7480:
7476:
7471:
7469:
7465:
7461:
7458:
7454:
7453:circumference
7449:
7447:
7443:
7439:
7434:
7432:
7413:
7408:
7404:
7400:
7397:
7394:
7391:
7384:
7383:
7382:
7380:
7376:
7373:
7369:
7365:
7355:
7353:
7348:
7346:
7336:
7332:
7330:
7314:
7302:
7297:
7290:
7283:
7280:
7275:
7270:
7267:
7259:
7253:
7247:
7242:
7238:
7234:
7231:
7228:
7225:
7218:
7216:
7213:
7212:
7208:
7204:
7190:
7187:
7184:
7181:
7176:
7172:
7168:
7165:
7162:
7155:
7153:
7150:
7149:
7145:
7141:
7120:
7116:
7112:
7107:
7103:
7097:
7094:
7088:
7085:
7082:
7079:
7072:
7070:
7066:
7063:
7062:
7058:
7054:
7037:
7034:
7031:
7025:
7022:
7019:
7016:
7013:
7006:
7004:
7000:
6997:
6996:
6992:
6988:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6944:
6941:
6938:
6931:
6929:
6926:
6925:
6921:
6917:
6901:
6897:
6893:
6890:
6885:
6881:
6877:
6874:
6871:
6868:
6861:
6858:
6855:
6854:
6851:
6849:
6848:
6844:
6842:
6841:
6837:
6833:
6819:
6816:
6810:
6804:
6798:
6795:
6784:
6778:
6773:
6768:
6764:
6760:
6757:
6750:
6748:
6745:
6744:
6740:
6736:
6720:
6717:
6714:
6711:
6708:
6701:
6699:
6696:
6695:
6691:
6687:
6669:
6665:
6662:
6659:
6653:
6648:
6644:
6638:
6635:
6630:
6627:
6620:
6618:
6615:
6614:
6610:
6606:
6604:
6602:
6587:
6584:
6581:
6578:
6575:
6551:
6545:
6541:
6537:
6531:
6526:
6522:
6518:
6515:
6512:
6505:
6503:
6500:
6499:
6496:
6494:
6479:
6476:
6467:
6452:
6449:
6423:
6420:
6411:
6408:
6405:
6402:
6393:
6390:
6381:
6378:
6375:
6372:
6366:
6363:
6340:
6331:
6328:
6319:
6316:
6310:
6307:
6301:
6298:
6289:
6270:
6267:
6264:
6261:
6250:
6246:
6244:
6223:
6219:
6216:
6206:
6203:
6198:
6194:
6190:
6184:
6181:
6175:
6148:
6145:
6136:
6133:
6128:
6124:
6116:
6113:
6109:
6103:
6076:
6073:
6064:
6061:
6056:
6052:
6048:
6045:
6018:
6015:
6006:
6003:
5998:
5994:
5990:
5984:
5981:
5975:
5951:
5947:
5944:
5938:
5933:
5929:
5926:
5920:
5917:
5914:
5907:
5905:
5891:
5881:
5878:
5877:
5873:
5869:
5851:
5847:
5838:
5833:
5830:
5824:
5821:
5818:
5811:
5809:
5805:
5804:
5800:
5796:
5778:
5774:
5768:
5761:
5758:
5753:
5750:
5743:
5741:
5737:
5736:
5732:
5728:
5710:
5706:
5700:
5697:
5694:
5685:
5682:
5675:
5673:
5670:
5669:
5665:
5661:
5643:
5639:
5635:
5632:
5629:
5622:
5620:
5619:Parallelogram
5617:
5616:
5612:
5608:
5594:
5591:
5586:
5583:
5578:
5575:
5568:
5566:
5562:
5559:
5558:
5554:
5550:
5532:
5528:
5522:
5518:
5512:
5509:
5502:
5500:
5498:
5492:
5488:
5487:
5483:
5479:
5461:
5457:
5453:
5448:
5444:
5440:
5433:
5430:
5425:
5422:
5415:
5413:
5410:
5409:
5391:
5388:
5385:
5382:
5379:
5370:
5367:
5361:
5358:
5350:
5346:
5325:
5322:
5319:
5310:
5307:
5304:
5295:
5292:
5289:
5283:
5278:
5275:
5268:
5266:
5264:
5258:
5255:
5254:
5250:
5246:
5227:
5221:
5218:
5215:
5212:
5207:
5204:
5199:
5196:
5189:
5187:
5184:
5183:
5179:
5175:
5159:
5156:
5151:
5148:
5143:
5140:
5133:
5131:
5128:
5127:
5123:
5119:
5105:
5102:
5099:
5096:
5089:
5087:
5084:
5083:
5079:
5075:
5059:
5055:
5051:
5048:
5041:
5039:
5036:
5035:
5031:
5028:
5025:
5024:
4999:
4996:
4993:
4989:
4986:
4981:
4974:
4955:
4949:
4929:
4923:
4919:
4915:
4912:
4905:
4904:
4903:
4887:
4877:
4874:
4871:
4865:
4862:
4859:
4814:
4808:
4805:
4802:
4791:
4778:
4759:
4756:
4753:
4749:
4746:
4740:
4737:
4732:
4727:
4721:
4713:
4704:
4699:
4694:
4689:
4683:
4675:
4666:
4657:
4653:
4649:
4646:
4639:
4638:
4637:
4623:
4601:
4591:
4588:
4585:
4579:
4576:
4573:
4550:
4544:
4541:
4538:
4532:
4529:
4526:
4509:
4505:
4489:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4439:
4431:
4428:
4425:
4421:
4417:
4399:
4395:
4392:
4386:
4383:
4375:
4372:
4369:
4365:
4361:
4345:
4342:
4339:
4336:
4333:
4325:
4322:
4319:
4303:
4300:
4280:
4277:
4274:
4266:
4262:
4243:
4240:
4237:
4231:
4228:
4225:
4217:
4214:
4211:
4193:
4189:
4185:
4177:
4174:
4159:
4156:
4153:
4131:
4127:
4123:
4115:
4111:
4090:
4087:
4084:
4078:
4075:
4055:
4052:
4049:
4046:
4041:
4037:
4033:
4025:
4021:
4004:
3996:
3992:
3988:
3983:
3979:
3973:
3970:
3966:
3962:
3959:
3951:
3948:
3947:
3941:
3927:
3924:
3921:
3917:
3912:
3904:
3901:
3898:
3890:
3887:
3882:
3874:
3870:
3866:
3859:
3855:
3851:
3843:
3840:
3837:
3834:
3829:
3825:
3815:
3812:
3804:
3800:
3796:
3792:
3788:
3769:
3766:
3763:
3754:
3751:
3748:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3719:
3715:
3711:
3708:
3702:
3696:
3693:
3687:
3681:
3673:
3660:
3656:
3652:
3636:
3633:
3630:
3623:
3615:
3607:
3598:
3588:
3584:
3576:
3572:
3567:
3561:
3558:
3549:
3548:
3546:
3542:
3526:
3523:
3513:
3510:
3504:
3501:
3495:
3492:
3486:
3476:
3472:
3464:
3460:
3455:
3449:
3446:
3441:
3438:
3435:
3428:
3425:
3419:
3412:
3408:
3400:
3396:
3391:
3387:
3384:
3381:
3378:
3371:
3368:
3362:
3355:
3351:
3343:
3339:
3334:
3326:
3325:
3324:
3323:
3319:
3298:
3294:
3281:
3275:
3267:
3263:
3250:
3221:
3215:
3212:
3206:
3200:
3194:
3188:
3176:
3166:
3162:
3161:
3146:
3143:
3140:
3134:
3130:
3126:
3121:
3118:
3113:
3110:
3103:
3102:
3098:
3095:expressed in
3079:
3073:
3070:
3067:
3059:
3058:
3039:
3033:
3013:
3010:
3007:
2997:
2991:
2988:
2982:
2976:
2968:
2963:
2959:
2955:
2952:
2945:
2944:
2940:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2904:
2903:
2888:
2885:
2882:
2875:
2869:
2864:
2859:
2855:
2851:
2848:
2841:
2840:
2836:
2832:
2828:
2824:
2820:
2819:
2814:
2807:
2803:
2799:
2795:
2790:
2778:
2774:
2770:
2754:
2751:
2746:
2743:
2738:
2735:
2727:
2723:
2719:
2716:
2712:
2700:
2688:
2676:
2655:
2651:
2645:
2641:
2637:
2632:
2628:
2622:
2618:
2614:
2609:
2605:
2599:
2595:
2591:
2586:
2582:
2576:
2572:
2568:
2563:
2559:
2553:
2549:
2545:
2540:
2536:
2530:
2526:
2516:
2513:
2501:
2495:
2489:
2469:
2463:
2460:
2457:
2454:
2448:
2445:
2419:
2416:
2413:
2410:
2407:
2398:
2395:
2389:
2386:
2378:
2374:
2370:
2349:
2346:
2343:
2334:
2331:
2328:
2319:
2316:
2313:
2307:
2298:can be used:
2297:
2296:
2291:
2287:
2283:
2279:
2263:
2260:
2254:
2251:
2240:
2236:
2235:
2217:
2213:
2210:
2207:
2200:
2197:
2188:
2174:
2172:
2167:
2157:
2153:
2149:
2148:
2147:
2145:
2144:
2139:
2135:
2131:
2126:
2124:
2120:
2116:
2112:
2108:
2100:
2096:
2092:
2088:
2084:
2079:
2055:
2052:
2049:
2046:
2043:
2040:
2033:
2032:
2031:
2028:
2022:
2017:
2013:
2009:
2003:
1979:
1974:
1970:
1966:
1962:
1958:
1955:
1947:
1943:
1939:
1934:
1930:
1922:
1917:
1914:
1910:
1906:
1902:
1898:
1891:
1890:
1889:
1887:
1883:
1879:
1875:
1870:
1867:
1861:
1852:
1848:
1844:
1843:
1842:
1839:
1832:
1826:
1825:circumference
1821:
1816:
1811:
1806:
1802:
1796:
1788:
1787:parallelogram
1784:
1779:
1765:
1763:
1759:
1739:
1736:
1731:
1728:
1723:
1720:
1713:
1712:
1711:
1709:
1705:
1701:
1692:
1684:
1680:
1676:
1675:
1674:
1672:
1668:
1664:
1663:parallelogram
1660:
1656:
1647:
1642:
1638:
1637:Triangle area
1628:
1626:
1622:
1618:
1614:
1610:
1606:
1597:
1593:
1589:
1588:
1587:
1580:
1576:
1566:
1562:
1558:
1557:
1556:
1546:
1533:
1524:
1522:
1519:
1515:
1511:
1507:
1476:
1472:
1466:
1463:
1460:
1456:
1452:
1447:
1444:
1441:
1437:
1431:
1427:
1418:
1415:
1412:
1407:
1404:
1401:
1397:
1384:
1381:
1376:
1373:
1366:
1365:
1364:
1362:
1358:
1355:
1352:-1) of whose
1351:
1347:
1326:
1322:
1318:
1313:
1309:
1298:
1294:
1288:
1276:Area formulas
1273:
1271:
1270:surface areas
1267:
1263:
1253:
1251:
1248:locations by
1247:
1243:
1239:
1235:
1225:
1223:
1219:
1215:
1211:
1208:
1207:quadrilateral
1204:
1200:
1196:
1183:
1179:
1175:
1171:
1169:
1168:
1163:
1159:
1155:
1151:
1150:mathematician
1147:
1142:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1109:
1101:Triangle area
1098:
1096:
1092:
1088:
1080:
1073:
1066:
1065:
1060:
1056:
1052:
1048:
1047:
1040:
1038:
1034:
1030:
1026:
1022:
1016:
998:
995:
992:
989:
986:
985:
984:
981:
979:
974:
972:
964:
963:
962:
961:, such that:
960:
955:
949:
948:
947:
945:
940:
938:
934:
930:
922:
921:
920:
918:
910:
909:
908:
906:
905:metric system
902:
896:
870:
868:square metres
857:
855:square metres
844:
841:
840:
839:
833:
830:
826:
825:
824:
818:
817:
816:
810:
806:
802:
801:
800:
798:
785:
781:
778:
774:
772:square metres
771:
767:
766:
765:
762:
759:
756:
749:
740:
738:
733:
731:
727:
723:
719:
715:
711:
710:square metres
707:
699:
694:
685:
678:
674:
670:
665:
661:
655:
651:
647:
643:
638:
633:
629:
625:
620:
616:
612:
608:
604:
600:
596:
591:
587:
583:
579:
575:
570:
566:
562:
558:
553:
549:
545:
541:
537:
533:
529:
524:
520:
516:
512:
508:
504:
500:
495:
490:
486:
480:
476:
471:
467:
463:
459:
454:
450:
446:
442:
438:
434:
430:
426:
422:
418:
412:
408:
402:
398:
393:
389:
385:
381:
376:
372:
367:
363:
359:
358:
357:
355:
349:
339:
336:
334:
330:
326:
322:
318:
314:
309:
307:
303:
299:
295:
290:
288:
284:
280:
276:
272:
268:
264:
260:
255:
253:
250:
249:dimensionless
246:
242:
238:
234:
230:
226:
221:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
178:
173:
172:planar lamina
169:
165:
161:
158:'s size on a
157:
153:
149:
127:
111:
109:
105:
102:
98:
96:
89:
86:
83:
81:
77:
74:
70:
67:
61:
57:are the same.
56:
50:
45:
40:
37:
33:
19:
9950:
9934:
9910:
9894:
9868:
9858:
9848:
9829:
9818:
9817:
9793:
9783:
9773:
9751:
9741:
9686:
9675:
9618:
9608:
9536:
9520:
9517:
9513:
9509:
9497:
9472:
9471:
9467:angular jerk
9451:
9422:
9421:
9390:
9386:acceleration
9361:
9360:
9350:
9329:
9328:
9316:
9304:
9287:
9276:
9261:
9251:
9231:
9194:
9193:
9183:
9163:
9158:
9150:displacement
9143:
9137:
9131:
9120:
9094:
9074:
9059:
9043:
9036:
8917:
8911:
8901:
8893:
8888:
8880:
8836:(1): 1–147.
8833:
8829:
8819:
8807:. Retrieved
8787:
8780:
8768:. Retrieved
8718:
8712:
8688:
8682:
8657:
8651:
8645:
8633:. Retrieved
8605:
8601:
8561:
8554:
8530:
8520:
8508:. Retrieved
8487:
8478:
8474:
8461:
8448:
8441:
8430:. Retrieved
8426:
8417:
8405:
8392:
8383:
8377:
8331:
8312:
8305:
8281:
8271:
8260:. Retrieved
8256:
8246:
8235:. Retrieved
8231:
8222:
8211:. Retrieved
8208:Housing News
8207:
8197:
8186:. Retrieved
8184:. 2020-08-04
8181:
8172:
8161:. Retrieved
8145:
8119:. Retrieved
8112:
8104:
8085:
8079:
8064:
8059:
8043:
8038:
8023:
8001:. Retrieved
7997:
7988:
7976:. Retrieved
7930:
7920:
7893:
7879:
7867:. Retrieved
7834:. Retrieved
7828:. Math.com.
7791:. Retrieved
7630:
7591:
7588:
7577:
7570:
7567:
7559:filling area
7556:
7552:soap bubbles
7545:
7542:Optimization
7533:
7530:
7511:
7488:
7472:
7467:
7463:
7459:
7450:
7437:
7435:
7428:
7378:
7374:
7367:
7361:
7349:
7342:
7328:
7068:
7002:
6847:Surface area
6845:
6567:
6493:circumcircle
6254:
5967:
5883:
5494:
5260:
4774:
4518:
4507:
4503:
4423:
4419:
4415:
4367:
4363:
4359:
4317:
4264:
4260:
4209:
4113:
4109:
4023:
4019:
3798:
3794:
3790:
3786:
3669:
3544:
2938:
2934:
2926:
2922:
2834:
2830:
2826:
2822:
2805:
2801:
2797:
2793:
2772:
2768:
2702:
2690:
2678:
2499:
2493:
2487:
2376:
2372:
2368:
2293:
2289:
2285:
2281:
2277:
2165:
2162:
2155:
2151:
2141:
2140:in his work
2127:
2107:surface area
2104:
2078:Surface area
2026:
2020:
2005:
1871:
1865:
1857:
1850:
1846:
1837:
1830:
1819:
1809:
1798:
1755:
1697:
1682:
1678:
1652:
1625:real numbers
1602:
1595:
1591:
1578:
1574:
1571:
1564:
1560:
1542:
1520:
1513:
1509:
1505:
1503:
1353:
1349:
1348:=0, 1, ...,
1345:
1290:
1259:
1231:
1192:
1173:
1165:
1143:
1138:
1130:
1122:
1113:
1078:
1071:
1062:
1045:
1041:
1018:
982:
975:
968:
956:
953:
941:
926:
914:
898:
837:
822:
814:
794:
763:
760:
757:
754:
734:
726:square miles
722:square yards
703:
683:
676:
672:
668:
663:
659:
653:
649:
645:
641:
636:
631:
627:
623:
618:
614:
610:
602:
598:
594:
589:
588:and breadth
585:
581:
577:
568:
564:
560:
556:
551:
547:
543:
539:
535:
531:
522:
518:
514:
510:
506:
502:
498:
493:
488:
484:
478:
474:
469:
465:
461:
452:
448:
444:
440:
436:
432:
428:
424:
420:
416:
410:
406:
400:
396:
394:then so are
391:
387:
383:
374:
370:
365:
361:
351:
337:
317:determinants
310:
298:surface area
291:
256:
233:square metre
222:
182:open surface
177:surface area
175:
163:
147:
146:
85:Square metre
80:SI unit
72:
68:
36:
9207:solid angle
7900:. pp.
7584:equilateral
1504:where when
1195:Brahmagupta
1182:Qin Jiushao
1167:Aryabhatiya
1009:Circle area
831:square feet
743:Conversions
718:square feet
550:is also in
414:, and also
252:real number
245:unit square
241:mathematics
9942:m s, N m s
9918:m s,
9864:Lagrangian
9789:Lagrangian
9029:Dimensions
9017:Dimensions
8809:1 February
8689:Matematika
8432:2023-07-12
8262:2023-09-20
8237:2023-09-20
8213:2023-09-20
8188:2023-09-20
8163:2008-02-13
8003:2018-10-09
7751:References
7733:Planimeter
7518:concurrent
5032:Variables
3659:planimeter
2138:Archimedes
2087:Archimedes
2016:semi-minor
2012:semi-major
1655:dissection
1617:arithmetic
1605:definition
1551:and width
1527:Rectangles
1154:astronomer
1148:, a great
1127:Archimedes
1051:Archimedes
1025:quadrature
978:South Asia
935:, and the
893:See also:
639:such that
346:See also:
267:rectangles
214:synecdoche
164:plane area
9958:ms,
9577:˙
9300:frequency
9227:frequency
8838:CiteSeerX
8674:124831432
8407:MathWorld
8346:π
7934:. Dover.
7602:π
7401:≤
7395:π
7372:perimeter
7239:∫
7235:π
7188:⋅
7182:⋅
7173:π
7086:π
7023:π
6894:π
6878:π
6817:≥
6765:∫
6715:π
6663:⋅
6636:θ
6538:π
6519:π
6421:π
6412:
6391:π
6382:
6329:π
6320:
6288:perimeter
6220:π
6207:
6146:π
6137:
6074:π
6065:
6016:π
6007:
5672:Trapezoid
5454:−
5323:−
5308:−
5293:−
5228:γ
5222:
5086:Rectangle
4972:∂
4962:∂
4956:×
4947:∂
4937:∂
4920:∬
4878:⊂
4872:∈
4719:∂
4711:∂
4681:∂
4673:∂
4654:∬
4592:⊂
4586:∈
4490:ℓ
4455:ℓ
4446:ℓ
4301:π
4278:π
4229:π
4154:π
4124:π
4076:π
4050:π
4034:π
3960:π
3925:≠
3905:α
3902:−
3899:β
3835:−
3770:β
3767:−
3755:α
3752:−
3694:−
3624:˙
3619:→
3608:×
3602:→
3568:∮
3514:˙
3502:−
3496:˙
3456:∮
3429:˙
3392:∮
3388:−
3372:˙
3335:∮
3285:→
3254:→
3180:→
3144:θ
3127:∫
3080:θ
2989:−
2960:∫
2856:∫
2752:−
2638:−
2615:−
2592:−
2464:
2347:−
2332:−
2317:−
2211:⋅
2047:π
1967:π
1940:−
1915:−
1911:∫
1758:trapezoid
1704:congruent
1702:into two
1667:trapezoid
1545:rectangle
1512:-1, then
1453:−
1416:−
1398:∑
1210:inscribed
1146:Aryabhata
1044:Euclid's
770:1,000,000
530:If a set
263:triangles
108:Dimension
9977:Category
9902:m s, N s
9725:J s
9721:m s
9696:m s
9657:m s
9628:m s
9604:momentum
9257:velocity
9127:position
9116:distance
9070:absement
8995:SI units
8946:Archived
8942:15049234
8934:27642581
8864:Archived
8803:Archived
8764:Archived
8734:Archived
8704:Archived
8626:Archived
8577:Archived
8501:Archived
8297:Archived
8279:(2003).
8154:Archived
8086:Calculus
8022:(1976).
7972:Archived
7928:(1959).
7863:Archived
7830:Archived
7787:Archived
7674:See also
7526:incircle
7522:centroid
7485:Fractals
7284:′
6999:Cylinder
6747:Integral
6601:diameter
6466:incircle
5806:Regular
5738:Regular
5491:triangle
5489:Regular
5257:Triangle
5186:Triangle
5130:Triangle
5029:Formula
4482:, where
4414:, where
4358:, where
4316:, where
4259:, where
4216:Cylinder
4208:, where
4018:, where
2919:function
2915:integral
2767:, where
2715:calculus
2239:triangle
2171:calculus
2117:(or any
2115:cylinder
2099:cylinder
1996:Ellipses
1874:calculus
1762:polygons
1708:triangle
1700:diagonal
1613:geometry
1357:vertices
1268:and the
1046:Elements
907:, with:
648:) ≤ c ≤
360:For all
329:analysis
313:geometry
283:calculus
259:formulas
186:boundary
174:, while
9930:rotatum
9709:actergy
9645:actergy
9614:impulse
9534::
9285::
9215:rad, sr
8860:0697984
8635:15 July
8622:2686282
8121:15 July
7869:15 July
7561:of the
7514:medians
7492:fractal
7479:apothem
7442:rhombus
6698:Ellipse
6495:radius
5904:sides)
5808:octagon
5740:hexagon
5561:Rhombus
4374:pyramid
3543:or the
2917:of one
2913:to the
2726:integer
2276:(where
2008:ellipse
1815:sectors
1783:sectors
1773:Circles
1518:modulus
1266:ellipse
1144:In 499
1131:Metrica
1123:Metrica
1091:hexagon
1027:of the
1004:History
917:hectare
730:squares
698:quadrat
666:, then
468:are in
390:are in
275:polygon
271:circles
225:squares
184:or the
160:surface
152:measure
150:is the
99:1
9844:energy
9825:moment
9813:torque
9769:energy
9747:weight
9719:
9701:action
9694:
9655:
9637:action
9626:
9597:
9544:
9516:⟩ = ∑
9480:
9431:
9370:
8940:
8932:
8858:
8840:
8795:
8770:6 July
8756:"Cone"
8726:
8696:
8672:
8620:
8569:
8542:
8324:
8289:
8092:
8071:
8050:
8030:
7978:3 July
7938:
7908:
7836:2 July
7793:3 July
7779:"Area"
7536:chords
7457:radius
7446:angles
7431:circle
7345:shapes
6928:Cuboid
6857:Sphere
6802:
6502:Circle
6468:radius
6274:
5038:Square
5026:Shape
4827:where
4563:where
4108:where
3785:where
3026:where
2907:graphs
2701:, and
2485:where
2367:where
2163:where
2130:sphere
2091:sphere
1801:circle
1669:and a
1293:simple
1246:vertex
1135:Euclid
937:myriad
933:hectad
931:, the
929:tetrad
809:inches
797:square
777:10,000
724:(yd),
720:(ft),
716:(km),
704:Every
580:is in
534:is in
492:is in
354:axioms
294:sphere
269:, and
243:, the
212:); by
206:volume
198:length
156:region
9946:power
9906:power
9876:m s,
9837:m s,
9801:m s,
9762:m s,
9737:force
9247:speed
9179:angle
8938:S2CID
8930:JSTOR
8670:S2CID
8629:(PDF)
8618:JSTOR
8598:(PDF)
8510:6 Feb
8504:(PDF)
8497:(PDF)
8471:(PDF)
8453:(PDF)
8157:(PDF)
8150:(PDF)
7902:45–61
7152:Torus
4324:Prism
2931:minus
2911:equal
2119:prism
2018:axes
1860:limit
1609:axiom
1250:Gauss
959:barns
873:2.589
860:0.836
847:0.092
807:= 12
688:Units
592:then
546:then
482:then
472:with
377:) ≥ 0
327:. In
237:metre
202:curve
200:of a
194:paint
188:of a
168:shape
154:of a
9983:Area
9890:yank
9854:work
9779:work
9493:mass
9447:jerk
9159:area
9090:time
9055:time
8811:2012
8793:ISBN
8772:2012
8724:ISBN
8694:ISBN
8637:2012
8567:ISBN
8540:ISBN
8512:2013
8322:ISBN
8287:ISBN
8123:2012
8090:ISBN
8069:ISBN
8048:ISBN
8028:ISBN
7980:2012
7936:ISBN
7906:ISBN
7871:2012
7838:2012
7795:2012
7362:The
7065:Cone
5565:Kite
4902:is:
4616:and
4176:Cube
3950:Cone
2825:and
2804:and
2497:and
2123:cone
2095:disk
2024:and
2014:and
1805:disk
1639:and
1216:and
1160:and
944:acre
942:The
805:foot
675:) =
662:and
617:and
609:Let
601:) =
563:) =
554:and
538:and
517:) −
509:) =
496:and
464:and
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435:) +
427:) =
404:and
386:and
148:Area
55:disk
42:Area
9924:MLT
9839:N m
9807:MLT
9760:kg
9665:MLT
9661:J s
9632:N s
9478:rad
9456:m s
9429:rad
9396:m s
9368:rad
9292:m s
9267:m s
9201:rad
9080:m s
8922:doi
8918:115
8848:doi
8662:doi
8610:doi
8536:306
6409:sin
6379:tan
6317:cot
6204:sin
6134:cot
6062:tan
6004:cot
5219:sin
4068:or
3099:is:
2929:),
2833:to
2461:sin
2154:= 4
1849:= π
1659:sum
1623:of
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1236:by
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976:In
901:are
881:336
878:110
875:988
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364:in
319:in
220:".
170:or
91:In
71:or
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6588::
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5701:c
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5686:=
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5587:2
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5579:=
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5563:/
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3442:=
3439:t
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3401:0
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3385:=
3382:t
3379:d
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3320::
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3195:=
3192:)
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3141:d
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3114:=
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3068:r
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3034:f
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2992:g
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2923:f
2889:.
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2879:)
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2870:f
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2852:=
2849:A
2835:b
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2806:b
2802:a
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2779:.
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2311:(
2308:s
2290:h
2286:B
2282:h
2278:B
2264:h
2261:B
2255:2
2252:1
2218:2
2214:h
2208:b
2201:=
2198:A
2166:r
2152:A
2056:.
2053:y
2050:x
2044:=
2041:A
2027:y
2021:x
1980:.
1975:2
1971:r
1963:=
1959:x
1956:d
1948:2
1944:x
1935:2
1931:r
1923:r
1918:r
1907:2
1903:=
1899:A
1866:r
1864:π
1851:r
1847:A
1838:r
1836:π
1831:r
1829:π
1820:r
1810:r
1789:.
1740:h
1737:b
1732:2
1729:1
1724:=
1721:A
1679:A
1596:s
1592:A
1584:s
1579:w
1575:l
1561:A
1553:w
1549:l
1539:.
1521:n
1514:i
1510:n
1508:=
1506:i
1487:|
1482:)
1477:i
1473:y
1467:1
1464:+
1461:i
1457:x
1448:1
1445:+
1442:i
1438:y
1432:i
1428:x
1424:(
1419:1
1413:n
1408:0
1405:=
1402:i
1392:|
1385:2
1382:1
1377:=
1374:A
1354:n
1350:n
1346:i
1344:(
1332:)
1327:i
1323:y
1319:,
1314:i
1310:x
1306:(
1184:.
1152:-
1110:.
1083:π
1079:r
1076:π
1072:r
1069:π
829:9
811:,
680:.
677:c
673:Q
671:(
669:a
664:T
660:S
656:)
654:T
652:(
650:a
646:S
644:(
642:a
637:c
632:T
628:Q
624:S
619:T
615:S
611:Q
606:.
599:R
597:(
595:a
590:k
586:h
582:M
578:R
573:.
571:)
569:T
567:(
565:a
561:S
559:(
557:a
552:M
548:T
544:T
540:S
536:M
532:S
527:.
525:)
523:S
521:(
519:a
515:T
513:(
511:a
507:S
505:−
503:T
501:(
499:a
494:M
489:S
485:T
479:T
475:S
470:M
466:T
462:S
457:.
455:)
453:T
449:S
447:(
445:a
441:T
439:(
437:a
433:S
431:(
429:a
425:T
423:∪
421:S
419:(
417:a
411:T
407:S
401:T
397:S
392:M
388:T
384:S
379:.
375:S
373:(
371:a
366:M
362:S
128:2
122:L
101:m
73:S
69:A
34:.
20:)
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