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This property can also be used to easily derive the formula for the area of a circle, because as the number of sides approaches infinity, the regular polygon's area approaches the area of the inscribed circle of radius
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259:, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height. The following formulations are all equivalent:
137:("that which is laid down"), indicating a generic line written down. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be
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is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the
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396:{\displaystyle A={\tfrac {1}{2}}nsa={\tfrac {1}{2}}pa={\tfrac {1}{4}}ns^{2}\cot {\frac {\pi }{n}}=na^{2}\tan {\frac {\pi }{n}}}
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according to the following formula, which also states that the area is equal to the apothem multiplied by half the
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to one of its sides. The word "apothem" can also refer to the length of that line segment and comes from the
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787:"德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?"
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414:. It is also the minimum distance between any side of the polygon and its center.
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615:{\displaystyle a={\frac {s}{2\tan {\frac {\pi }{n}}}}=R\cos {\frac {\pi }{n}}.}
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Segment from the center of a polygon to the midpoint of one of its sides
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506:{\displaystyle A={\frac {pa}{2}}={\frac {(2\pi r)r}{2}}=\pi r^{2}}
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695:{\displaystyle a={\frac {s}{2}}\tan {\frac {\pi (n-2)}{2n}}.}
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The apothem of a regular polygon can be found multiple ways.
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These formulae can still be used even if only the perimeter
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235:{\displaystyle A={\frac {nsa}{2}}={\frac {pa}{2}}.}
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406:An apothem of a regular polygon will always be a
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245:This formula can be derived by partitioning the
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848:Apothem of pyramid or truncated pyramid
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868:The Wolfram Demonstrations Project
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757:Circumradius of a regular polygon
625:The apothem can also be found by
129:("put away, put aside"), made of
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532:-sided polygon with side length
89:. The green line shows the case
797:from the original on 2015-09-19
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161:-sided polygon of side length
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864:"Sagitta, Apothem, and Chord"
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842:Apothem of a regular polygon
87:rectangle with the same area
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844:With interactive animation
106:(sometimes abbreviated as
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709:and the number of sides
153:can be used to find the
19:Not to be confused with
820:www.merriam-webster.com
816:"Definition of APOTHEM"
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145:Properties of apothems
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791:University of Hawaii
752:Chord (trigonometry)
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785:Shaneyfelt, Ted V.
517:Finding the apothem
257:isosceles triangles
853:2021-04-21 at the
762:Sagitta (geometry)
713:are known because
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124:ancient Greek
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120:perpendicular
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30:Apothem of a
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823:. Retrieved
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799:. Retrieved
780:
767:Slant height
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714:
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706:
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624:
540:
538:circumradius
533:
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524:The apothem
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149:The apothem
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77:1, with the
75:circumradius
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62:
54:
50:
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825:2022-02-17
801:2015-11-08
773:References
73:sides and
41:Graphs of
21:Apophthegm
670:−
661:π
655:
602:π
597:
577:π
572:
491:π
470:π
386:π
381:
354:π
349:
254:congruent
167:perimeter
139:congruent
885:Polygons
879:Category
851:Archived
795:Archived
746:See also
118:that is
81:,
61:,
53:,
45:,
735:
719:
410:of the
127:ἀπόθεμα
116:polygon
110:) of a
104:apothem
51:apothem
32:hexagon
408:radius
169:since
57:; and
741:Notes
536:, or
85:of a
155:area
135:θέμα
102:The
79:base
59:area
43:side
652:tan
594:cos
569:tan
378:tan
346:cot
131:ἀπό
108:apo
95:= 6
69:of
65:of
881::
866:.
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171:ns
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870:.
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727:/
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715:s
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690:.
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681:2
676:)
673:2
667:n
664:(
647:2
644:s
639:=
636:a
610:.
605:n
591:R
588:=
580:n
566:2
562:s
557:=
554:a
541:R
534:s
530:n
526:a
499:2
495:r
488:=
483:2
479:r
476:)
473:r
467:2
464:(
458:=
453:2
449:a
446:p
440:=
437:A
424:a
420:r
389:n
373:2
369:a
365:n
362:=
357:n
341:2
337:s
333:n
327:4
324:1
318:=
315:a
312:p
306:2
303:1
297:=
294:a
291:s
288:n
282:2
279:1
273:=
270:A
251:n
247:n
230:.
225:2
221:a
218:p
212:=
207:2
203:a
200:s
197:n
191:=
188:A
175:p
163:s
159:n
151:a
98:.
93:n
83:b
71:n
63:A
55:a
47:s
23:.
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