Editing Regular polygon (section)

==Regular convex polygons==
All regular [[simple polygon]]s (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also [[Similarity (geometry)|similar]].

An ''n''-sided convex regular polygon is denoted by its [[Schläfli symbol]] <math>\{n\}</math>. For <math>n<3</math>, we have two [[degeneracy (mathematics)|degenerate]] cases:
; [[Monogon]] {1}: Degenerate in [[Euclidean geometry|ordinary space]]. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any [[abstract polytope|abstract polygon]].)
; [[Digon]] {2}; a "double line segment": Degenerate in [[Euclidean geometry|ordinary space]]. (Some authorities{{weasel inline|date=December 2024}} do not regard the digon as a true polygon because of this.)

In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of [[uniform polyhedra]] must be regular and the faces will be described simply as triangle, square, pentagon, etc.
[[File:annuli_with_same_area_around_unit_regular_polygons.svg|class=skin-invert-image|thumb|upright=0.8|As a corollary of the [[annulus (mathematics)|annulus]] chord formula, the&nbsp;area bounded by the [[circumcircle]] and [[incircle]] of every unit convex regular polygon is {{pi}}/4]]

===Angles===
For a regular convex ''n''-gon, each [[interior angle]] has a measure of:
: <math>\frac{180(n - 2)}{n}</math> degrees;
: <math>\frac{(n - 2)\pi}{n}</math> radians; or
: <math>\frac{(n - 2)}{2n}</math> full [[turn (geometry)|turns]],

and each [[exterior angle]] (i.e., [[supplementary angle|supplementary]] to the interior angle) has a measure of <math>\tfrac{360}{n}</math> degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

As ''n'' approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a [[myriagon]]) the internal angle is 179.964°. As the number of sides increases, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line (see [[apeirogon]]). For this reason, a circle is not a polygon with an infinite number of sides.

===Diagonals===
For <math>n>2</math>, the number of [[diagonal]]s is <math>\tfrac{1}{2}n(n - 3)</math>; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces.{{efn|{{oeis|A007678}}}}

For a regular ''n''-gon inscribed in a circle of radius <math>1</math>, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals ''n''.

===Points in the plane===

For a regular simple {{mvar|n}}-gon with [[circumradius]] {{mvar|R}} and distances {{mvar|d<sub>i</sub>}} from an arbitrary point in the plane to the vertices, we have<ref>Park, Poo-Sung. "Regular polytope distances", [[Forum Geometricorum]] 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf</ref>

:<math>\frac{1}{n}\sum_{i=1}^n d_i^4 + 3R^4 = \biggl(\frac{1}{n}\sum_{i=1}^n d_i^2 + R^2\biggr)^2.</math>

For higher powers of distances <math>d_i</math> from an arbitrary point in the plane to the vertices of a regular {{mvar|n}}-gon, if

:<math>S^{(2m)}_{n}=\frac 1n\sum_{i=1}^n d_i^{2m}</math>,

then<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 24 February 2025|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

:<math>S^{(2m)}_{n} = \left(S^{(2)}_{n}\right)^m + \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\binom{m}{2k}\binom{2k}{k}R^{2k}\left(S^{(2)}_{n} - R^2\right)^k\left(S^{(2)}_{n}\right)^{m-2k}</math>,

and

:<math> S^{(2m)}_{n} = \left(S^{(2)}_{n}\right)^m + \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\frac{1}{2^k}\binom{m}{2k}\binom{2k}{k} \left(S^{(4)}_{n} -\left(S^{(2)}_{n}\right)^2\right)^k\left(S^{(2)}_{n}\right)^{m-2k}</math>,

where {{mvar|m}} is a positive integer less than {{mvar|n}}.

If {{mvar|L}} is the distance from an arbitrary point in the plane to the centroid of a regular {{mvar|n}}-gon with circumradius {{mvar|R}}, then<ref name= Mamuka />

:<math>\sum_{i=1}^n d_i^{2m}=n\left(\left(R^2+L^2\right)^m+ \sum_{k=1}^{\left\lfloor m/2\right\rfloor}\binom{m}{2k}\binom{2k}{k}R^{2k}L^{2k}\left(R^2+L^2\right)^{m-2k}\right)</math>,
where <math>m = 1, 2, \dots, n - 1</math>.

====Interior points====
For a regular {{mvar|n}}-gon, the sum of the perpendicular distances from any interior point to the {{mvar|n}} sides is {{mvar|n}} times the [[apothem]]<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).</ref>{{rp|p. 72}} (the apothem being the distance from the center to any side). This is a generalization of [[Viviani's theorem]] for the ''n'' = 3 case.<ref>Pickover, Clifford A, ''The Math Book'', Sterling, 2009: p. 150</ref><ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>

===Circumradius===
[[Image:PolygonParameters.png|class=skin-invert-image|thumb|left|180px|Regular [[pentagon]] (''n'' = 5) with [[edge (geometry)|side]] ''s'', [[circumradius]] ''R'' and [[apothem]] ''a'']]
<div class="skin-invert-image">{{regular polygon side count graph.svg}}</div>
The [[circumradius]] ''R'' from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the [[apothem]] ''a'' by
:<math>R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} = \frac{a}{\cos\left(\frac{\pi}{n} \right)} \quad_,\quad a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}</math>

For [[constructible polygon]]s, [[algebraic expression]]s for these relationships exist {{xref|(see {{slink|Bicentric polygon|Regular polygons}})}}.

The sum of the perpendiculars from a regular ''n''-gon's vertices to any line tangent to the circumcircle equals ''n'' times the circumradius.<ref name=Johnson/>{{rp|p. 73}}

The sum of the squared distances from the vertices of a regular ''n''-gon to any point on its circumcircle equals 2''nR''<sup>2</sup> where ''R'' is the circumradius.<ref name=Johnson/>{{rp|p. 73}}

The sum of the squared distances from the midpoints of the sides of a regular ''n''-gon to any point on the circumcircle is 2''nR''<sup>2</sup> − {{sfrac|1|4}}''ns''<sup>2</sup>, where ''s'' is the side length and ''R'' is the circumradius.<ref name=Johnson/>{{rp|p. 73}}
{{Clear|left}}

If <math>d_i</math> are the distances from the vertices of a regular <math>n</math>-gon to any point on its circumcircle, then <ref name= Mamuka />

:<math>3\biggl(\sum_{i=1}^n d_i^2\biggr)^2 = 2n \sum_{i=1}^n d_i^4 </math>.

=== Dissections===
[[Coxeter]] states that every [[zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into <math>\tbinom{m}{2}</math> or {{nowrap|{{sfrac|1|2}}''m''(''m'' − 1)}} parallelograms.
These tilings are contained as subsets of vertices, edges and faces in orthogonal projections [[Hypercube|''m''-cubes]].<ref>[[Coxeter]], Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi.
The list {{OEIS2C|1=A006245}} gives the number of solutions for smaller polygons.
{| class="wikitable skin-invert-image"
|+ Example dissections for select even-sided regular polygons
!
|[[File:6-gon rhombic dissection.svg  |100px]]
|[[File:8-gon rhombic dissection.svg  |100px]]
|[[File:Sun_decagon.svg               |100px]]
|[[File:12-gon rhombic dissection.svg |100px]]
|[[File:14-gon-dissection-star.svg    |100px]]
|[[File:16-gon rhombic dissection.svg |100px]]
|- align=center valign=top
!scope=row style="text-align:left" |Sides
|[[Hexagon#Dissection|6]]
|[[Octagon#Dissection|8]]
|[[Decagon#Dissection|10]]
|[[Dodecagon#Dissection|12]]
|[[Tetradecagon#Dissection|14]]
|[[Hexadecagon#Dissection|16]]
|- align=center valign=top
!scope=row style="text-align:left" |Rhombs
|3
|6
|10
|15
|21
|28
|- 
|}
{| class="wikitable skin-invert-image"
!
|[[File:18-gon-dissection-star.svg    |100px]]
|[[File:20-gon rhombic dissection.svg |100px]]
|[[File:24-gon rhombic dissection.svg |100px]]
|[[File:30-gon-dissection-star.svg    |100px]]
|[[File:40-gon rhombic dissection.svg |100px]]
|[[File:50-gon-dissection-star.svg    |100px]]
|- align=center valign=top
!scope=row style="text-align:left" |Sides
|[[Octadecagon#Dissection|18]]
|[[Icosagon#Dissection|20]]
|[[Icositetragon#Dissection|24]]
|[[Triacontagon#Dissection|30]]
|40
|50
|- align=center valign=top
!scope=row style="text-align:left" |Rhombs
|36
|45
|66
|105
|190
|300
|}

===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>