Editing Regular polygon (section)

===Area===<!--This section is linked from [[Truncated icosahedron]]-->
The area ''A'' of a convex regular ''n''-sided polygon having [[Edge (geometry)|side]] ''s'', [[circumscribed circle|circumradius]] ''R'', [[apothem]] ''a'', and [[perimeter]] ''p'' is given by<ref>{{cite web |url=http://www.mathopenref.com/polygonregulararea.html |title=Math Open Reference |access-date=4 Feb 2014}}</ref><ref>{{cite web |url=http://www.mathwords.com/a/area_regular_polygon.htm |title=Mathwords}}</ref>
<math display="block">\begin{align}
A &= \tfrac{1}{2}nsa \\
  &= \tfrac{1}{2}pa \\
  &= \tfrac{1}{4}ns^2\cot\left(\tfrac{\pi}{n}\right) \\
  &= na^2\tan\left(\tfrac{\pi}{n}\right) \\
  &= \tfrac{1}{2}nR^2\sin\left(\tfrac{2\pi}{n}\right)
\end{align}</math>

For regular polygons with side ''s'' = 1, circumradius ''R'' = 1, or apothem ''a'' = 1, this produces the following table:{{efn|1=Results for ''R'' = 1 and ''a'' = 1 obtained with [[Maple (software)|Maple]], using function definition:
<syntaxhighlight lang="maple">
f := proc (n)
options operator, arrow;
[
 [convert(1/4*n*cot(Pi/n), radical), convert(1/4*n*cot(Pi/n), float)],
 [convert(1/2*n*sin(2*Pi/n), radical), convert(1/2*n*sin(2*Pi/n), float), convert(1/2*n*sin(2*Pi/n)/Pi, float)],
 [convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]
]
end proc
</syntaxhighlight>The expressions for ''n'' = 16 are obtained by twice applying the [[tangent half-angle formula]] to tan(π/4)}} ([[trigonometric functions|Since <math>\scriptstyle \cot x \rightarrow 1/x</math> as <math>\scriptstyle x \rightarrow 0</math>]], the area when <math>\scriptstyle s = 1</math> tends to <math>\scriptstyle n^2/4\pi</math> as <math>\scriptstyle n</math> grows large.)
{{Clear}}
<div style="overflow:auto">
{| class=wikitable style="text-align:center;"
|-
!                            rowspan="2" {{verth|Number<br/>of sides}}
! style="background:#ff9bac" colspan="2" | Area when side ''s'' = 1
! style="background:#6dd7af" colspan="3" | Area when circumradius ''R'' = 1
! style="background:#83c6ff" colspan="3" | Area when apothem ''a'' = 1
|-
! Exact
! Approxi{{shy}}mation
! Exact
! Approxi{{shy}}mation
! Relative to circum{{shy}}circle area
! Exact
! Approxi{{shy}}mation
! Relative to in{{shy}}circle area
|-
| ''n''
| <math>\scriptstyle \tfrac{n}{4}\cot\left(\tfrac{\pi}{n}\right)</math>  ||
| <math>\scriptstyle \tfrac{n}{2}\sin\left(\tfrac{2\pi}{n}\right)</math> || || <math>\scriptstyle \tfrac{n}{2\pi}\sin\left(\tfrac{2\pi}{n}\right)</math>
| <math>\scriptstyle n \tan\left(\tfrac{\pi}{n}\right)</math>            || || <math>\scriptstyle \tfrac{n}{\pi}\tan\left(\tfrac{\pi}{n}\right)</math>
|-
| [[Equilateral triangle|3]]
| {{tmath|\scriptstyle \tfrac{ \sqrt{3} }{4} }} || 0.433012702
| {{tmath|\scriptstyle \tfrac{3\sqrt{3} }{4} }} || 1.299038105 || 0.4134966714
| {{tmath|\scriptstyle        3\sqrt{3}      }} || 5.196152424 || 1.653986686
|-
| [[Square|4]]
| 1 || 1.000000000
| 2 || 2.000000000 || 0.6366197722
| 4 || 4.000000000 || 1.273239544
|-
| [[Regular pentagon|5]]
| {{tmath|\scriptstyle \tfrac{1}{4}\sqrt{25 + 10\sqrt{5} } }}                      || 1.720477401
| {{tmath|\scriptstyle \tfrac{5}{4}\sqrt{\tfrac{1}{2}\left(5 + \sqrt{5}\right)} }} || 2.377641291 || 0.7568267288
| {{tmath|\scriptstyle 5\sqrt{5 - 2\sqrt{5} } }}                                   || 3.632712640 || 1.156328347
|-
| [[Regular hexagon|6]]
| {{tmath|\scriptstyle \tfrac{3\sqrt{3} }{2} }} || 2.598076211
| {{tmath|\scriptstyle \tfrac{3\sqrt{3} }{2} }} || 2.598076211 || 0.8269933428
| {{tmath|\scriptstyle        2\sqrt{3}      }} || 3.464101616 || 1.102657791
|-
| [[Regular heptagon|7]]
|  || 3.633912444
|  || 2.736410189 || 0.8710264157
|  || 3.371022333 || 1.073029735
|-
| [[Regular octagon|8]]
| {{tmath|\scriptstyle   2 + 2\sqrt{2}           }} || 4.828427125
| {{tmath|\scriptstyle       2\sqrt{2}           }} || 2.828427125 || 0.9003163160
| {{tmath|\scriptstyle 8\left(\sqrt{2} - 1\right)}} || 3.313708500 || 1.054786175
|-
| [[Regular enneagon|9]]
|  || 6.181824194
|  || 2.892544244 || 0.9207254290
|  || 3.275732109 || 1.042697914
|-
| [[Regular decagon|10]]
| {{tmath|\scriptstyle \tfrac{5}{2}\sqrt{5 + 2\sqrt{5} } }}                        || 7.694208843
| {{tmath|\scriptstyle \tfrac{5}{2}\sqrt{\tfrac{1}{2}\left(5 - \sqrt{5}\right)} }} || 2.938926262 || 0.9354892840
| {{tmath|\scriptstyle 2\sqrt{25 - 10\sqrt{5} } }}                                 || 3.249196963 || 1.034251515
|-
| [[Regular hendecagon|11]]
|  || 9.365639907
|  || 2.973524496 || 0.9465022440
|  || 3.229891423 || 1.028106371
|-
| [[Regular dodecagon|12]]
| {{tmath|\scriptstyle 6 + 3\sqrt{3} }}               || 11.19615242
| 3                                      || 3.000000000 || 0.9549296586
| {{tmath|\scriptstyle 12\left(2 - \sqrt{3} \right)}} || 3.215390309 || 1.023490523
|-
| [[Regular tridecagon|13]]
|  || 13.18576833
|  || 3.020700617 || 0.9615188694
|  || 3.204212220 || 1.019932427
|-
| [[Regular tetradecagon|14]]
|  || 15.33450194
|  || 3.037186175 || 0.9667663859
|  || 3.195408642 || 1.017130161
|-
| [[Regular pentadecagon|15]]
| {{tmath|\scriptstyle  \tfrac{15}{8}\left(\sqrt{15} + \sqrt{3} + \sqrt{2\left(5 + \sqrt{5} \right)} \right)}}     || 17.64236291
| {{tmath|\scriptstyle  \tfrac{15}{16}\left(\sqrt{15} + \sqrt{3} - \sqrt{10 - 2\sqrt{5} } \right)}}                || 3.050524822 || 0.9710122088
| {{tmath|\scriptstyle  \tfrac{15}{2}\left(3\sqrt{3} - \sqrt{15} - \sqrt{2\left(25 - 11\sqrt{5} \right)} \right)}} || 3.188348426 || 1.014882824
|-
| [[Regular hexadecagon|16]]
| {{tmath|\scriptstyle  4 \left(1 + \sqrt{2} + \sqrt{2 \left(2 + \sqrt{2} \right)} \right)}}               || 20.10935797
| {{tmath|\scriptstyle  4\sqrt{2 - \sqrt{2} } }}                                                                      || 3.061467460 || 0.9744953584
| {{tmath|\scriptstyle  16 \left(1 + \sqrt{2}\right)\left(\sqrt{2 \left(2 - \sqrt{2} \right)} - 1\right)}} || 3.182597878 || 1.013052368
|-
| [[Regular heptadecagon|17]]
|  || 22.73549190
|  || 3.070554163 || 0.9773877456
|  || 3.177850752 || 1.011541311
|-
| [[Regular octadecagon|18]]
|  || 25.52076819
|  || 3.078181290 || 0.9798155361
|  || 3.173885653 || 1.010279181
|-
| [[Regular enneadecagon|19]]
|  || 28.46518943
|  || 3.084644958 || 0.9818729854
|  || 3.170539238 || 1.009213984
|-
| [[Regular icosagon|20]]
| {{tmath|\scriptstyle  5 \left(1 + \sqrt{5} + \sqrt{5 + 2\sqrt{5} } \right) }}    || 31.56875757
| {{tmath|\scriptstyle  \tfrac{5}{2}\left(\sqrt{5} - 1\right)}}                 || 3.090169944 || 0.9836316430
| {{tmath|\scriptstyle  20 \left(1 + \sqrt{5} -\sqrt{5 + 2\sqrt{5} } \right) }} || 3.167688806 || 1.008306663
|-
| 100
|  || 795.5128988
|  || 3.139525977 || 0.9993421565
|  || 3.142626605 || 1.000329117
|-
| [[Regular chiliagon|1000]]
|  || 79577.20975
|  || 3.141571983 || 0.9999934200
|  || 3.141602989 || 1.000003290
|-
| [[Regular myriagon|{{10^|4}}]]
|  || 7957746.893
|  || 3.141592448 || 0.9999999345
|  || 3.141592757 || 1.000000033
|-
| [[Regular megagon|{{10^|6}}]]
|  || 79577471545
|  || 3.141592654 || 1.000000000
|  || 3.141592654 || 1.000000000
|}
</div>

[[File:Polygons comparison.png|thumb|right|400px|Comparison of sizes of regular polygons with the same edge length, from [[equilateral triangle|three]] to [[hexacontagon|sixty]] sides. The size increases without bound as the number of sides approaches infinity.]]
Of all ''n''-gons with a given perimeter, the one with the largest area is regular.<ref>Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>