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Regular polygon

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From Wikipedia, the free encyclopedia

Equiangular and equilateral polygon

Regular polygon

Edges andvertices

n {\displaystyle n}

Schläfli symbol

\{n\}

Coxeter–Dynkin diagram

Symmetry group

D_n, order 2n

Dual polygon

Self-dual

Area
(with side length

s {\displaystyle s}

)

A={\tfrac {1}{4}}ns^{2}\cot \left({\frac {\pi }{n}}\right)

Internal angle

(n-2)\times {\frac {\pi }{n}}

Internal angle sum

\left(n-2\right)\times {\pi }

Inscribed circle diameter

d_{\text{IC}}=s\cot \left({\frac {\pi }{n}}\right)

Circumscribed circle diameter

d_{\text{OC}}=s\csc \left({\frac {\pi }{n}}\right)

Properties

Convex,cyclic,equilateral,isogonal,isotoxal

InEuclidean geometry, aregular polygon is apolygon that isdirect equiangular (all angles are equal in measure) andequilateral (all sides have the same length). Regular polygons may be eitherconvex orstar. In thelimit, a sequence of regular polygons with an increasing number of sides approximates acircle, if theperimeter orarea is fixed, or a regularapeirogon (effectively astraight line), if the edge length is fixed.

General properties

[edit]

Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols

These properties apply to all regular polygons, whether convex orstar:

A regularn-sided polygon hasrotational symmetry of ordern.

All vertices of a regular polygon lie on a common circle (thecircumscribed circle); i.e., they are concyclic points. That is, a regular polygon is acyclic polygon.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle orincircle that is tangent to every side at the midpoint. Thus a regular polygon is atangential polygon.

A regularn-sided polygon can be constructed withcompass and straightedge if and only if theodd prime factors ofn are distinctFermat primes.(Seeconstructible polygon.)

A regularn-sided polygon can be constructed withorigami if and only if $n=2^{a}3^{b}p_{1}\cdots p_{r}$ for some $r\in \mathbb {N}$ , where each distinct $p_{i}$ is aPierpont prime.^[1]

Symmetry

[edit]

Thesymmetry group of ann-sided regular polygon is thedihedral group D_n (of order 2n): D₂,D₃, D₄, ... It consists of the rotations in C_n, together withreflection symmetry inn axes that pass through the center. Ifn is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. Ifn is odd then all axes pass through a vertex and the midpoint of the opposite side.

Regular convex polygons

[edit]

All regularsimple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are alsosimilar.

Ann-sided convex regular polygon is denoted by itsSchläfli symbol $\{n\}$ . For $n<3$ , we have twodegenerate cases:

Monogon {1}; point: Degenerate inordinary 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 anyabstract polygon.)
Digon {2}; line segment: Degenerate inordinary space. (Some authorities^{[weasel words]} 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 ofuniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc.

As a corollary of theannulus chord formula, the area bounded by thecircumcircle andincircle of every unit convex regular polygon isπ/4

Angles

[edit]

For a regular convexn-gon, eachinterior angle has a measure of:

{\frac {(n-2)180}{n}}

degrees;

{\frac {(n-2)\pi }{n}}

radians; or

{\frac {(n-2)}{2n}}={\frac {1}{2}}-{\frac {1}{n}}

fullturns,

and eachexterior angle (i.e.,supplementary to the interior angle) has a measure of ${\tfrac {360}{n}}$ degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

Asn approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (amyriagon) 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 (seeapeirogon). For this reason, a circle is not a polygon with an infinite number of sides.

Diagonals

[edit]

For $n>2$ , the number ofdiagonals is ${\tfrac {1}{2}}n(n-3)$ ; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces.^[a]

For a regularn-gon inscribed in a circle of radius $1 {\displaystyle 1}$ , the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equalsn.

Points in the plane

[edit]

For a regular simplen-gon withcircumradiusR and distancesd_i from an arbitrary point in the plane to the vertices, we have^[2]

{\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}.

For higher powers of distances $d_{i}$ from an arbitrary point in the plane to the vertices of a regularn-gon, if

S_{n}^{(2m)}={\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2m}

then^[3]

S_{n}^{(2m)}=\left(S_{n}^{(2)}\right)^{m}+\sum _{k=1}^{\left\lfloor m/2\right\rfloor }{\binom {m}{2k}}{\binom {2k}{k}}R^{2k}\left(S_{n}^{(2)}-R^{2}\right)^{k}\left(S_{n}^{(2)}\right)^{m-2k}

and

S_{n}^{(2m)}=\left(S_{n}^{(2)}\right)^{m}+\sum _{k=1}^{\left\lfloor m/2\right\rfloor }{\frac {1}{2^{k}}}{\binom {m}{2k}}{\binom {2k}{k}}\left(S_{n}^{(4)}-\left(S_{n}^{(2)}\right)^{2}\right)^{k}\left(S_{n}^{(2)}\right)^{m-2k}

wherem is a positive integer less thann.

IfL is the distance from an arbitrary point in the plane to the centroid of a regularn-gon with circumradiusR, then^[3]

\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)

where $m=1,2,\dots ,n-1$ .

Interior points

[edit]

For a regularn-gon, the sum of the perpendicular distances from any interior point to then sides isn times theapothem^[4]^{: p. 72} (the apothem being the distance from the center to any side). This is a generalization ofViviani's theorem for then = 3 case.^[5]^[6]

Circumradius

[edit]

Graphs ofside, s;apothem, a; andarea, A ofregular polygons ofn sides andcircumradius 1, with thebase, b of arectangle with the same area. The green line shows the casen = 6.

ThecircumradiusR from the center of a regular polygon to one of the vertices is related to the side lengths or to theapothema by

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)}}

Forconstructible polygons,algebraic expressions for these relationships exist(seeBicentric polygon § Regular polygons).

The sum of the perpendiculars from a regularn-gon's vertices to any line tangent to the circumcircle equalsn times the circumradius.^[4]^{: p. 73}

The sum of the squared distances from the vertices of a regularn-gon to any point on its circumcircle equals 2nR² whereR is the circumradius.^[4]^{: p. 73}

The sum of the squared distances from the midpoints of the sides of a regularn-gon to any point on the circumcircle is 2nR² −⁠1/4⁠ns², wheres is the side length andR is the circumradius.^[4]^{: p. 73}

If $d_{i}$ are the distances from the vertices of a regular $n {\displaystyle n}$ -gon to any point on its circumcircle, then^[3]

3{\biggl (}\sum _{i=1}^{n}d_{i}^{2}{\biggr )}^{2}=2n\sum _{i=1}^{n}d_{i}^{4}

Dissections

[edit]

Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into ${\tbinom {m}{2}}$ or⁠1/2⁠m(m − 1) parallelograms.These tilings are contained as subsets of vertices, edges and faces in orthogonal projectionsm-cubes.^[7]

In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi. Regular polygons with 4m+2 sides can be dissected in a way with (2m+1)-fold radial symmetry.The listOEIS: A006245 gives the number of solutions for smaller polygons.

Examples of dissections for selected even-sided regular polygons

Sides	6	8	10	12	14	16
Rhombs	3	6	10	15	21	28


Sides	18	20	24	30	40	50
Rhombs	36	45	66	105	190	300

Area

[edit]

The areaA of a convex regularn-sided polygon havingsides,circumradiusR,apothema, andperimeterp is given by^[8]^[9] ${\begin{aligned}A&={\tfrac {1}{2}}nsa\\&={\tfrac {1}{2}}pa\\&={\tfrac {1}{4}}ns^{2}\cot \left({\tfrac {\pi }{n}}\right)\\&=na^{2}\tan \left({\tfrac {\pi }{n}}\right)\\&={\tfrac {1}{2}}nR^{2}\sin \left({\tfrac {2\pi }{n}}\right)\end{aligned}}$

For regular polygons with sides = 1, circumradiusR = 1, or apothema = 1, this produces the following table:^[b] (Since $\cot x\rightarrow 1/x$ as $x\rightarrow 0$ , the area when $s=1$ tends to $n^{2}/4\pi$ as $n {\displaystyle n}$ grows large.)

Number of sides	Area when sides = 1		Area when circumradiusR = 1			Area when apothema = 1
Number of sides	Exact	Approximation	Exact	Approximation	Relative to circumcircle area	Exact	Approximation	Relative to incircle area
n	$\scriptstyle {\tfrac {n}{4}}\cot \left({\tfrac {\pi }{n}}\right)$		$\scriptstyle {\tfrac {n}{2}}\sin \left({\tfrac {2\pi }{n}}\right)$		$\scriptstyle {\tfrac {n}{2\pi }}\sin \left({\tfrac {2\pi }{n}}\right)$	$\scriptstyle n\tan \left({\tfrac {\pi }{n}}\right)$		$\scriptstyle {\tfrac {n}{\pi }}\tan \left({\tfrac {\pi }{n}}\right)$
3	⁠ $\scriptstyle {\tfrac {\sqrt {3}}{4}}$ ⁠	0.433012702	⁠ $\scriptstyle {\tfrac {3{\sqrt {3}}}{4}}$ ⁠	1.299038105	0.4134966714	⁠ $\scriptstyle 3{\sqrt {3}}$ ⁠	5.196152424	1.653986686
4	1	1.000000000	2	2.000000000	0.6366197722	4	4.000000000	1.273239544
5	⁠ $\scriptstyle {\tfrac {1}{4}}{\sqrt {25+10{\sqrt {5}}}}$ ⁠	1.720477401	⁠ $\scriptstyle {\tfrac {5}{4}}{\sqrt {{\tfrac {1}{2}}\left(5+{\sqrt {5}}\right)}}$ ⁠	2.377641291	0.7568267288	⁠ $\scriptstyle 5{\sqrt {5-2{\sqrt {5}}}}$ ⁠	3.632712640	1.156328347
6	⁠ $\scriptstyle {\tfrac {3{\sqrt {3}}}{2}}$ ⁠	2.598076211	⁠ $\scriptstyle {\tfrac {3{\sqrt {3}}}{2}}$ ⁠	2.598076211	0.8269933428	⁠ $\scriptstyle 2{\sqrt {3}}$ ⁠	3.464101616	1.102657791
7		3.633912444		2.736410189	0.8710264157		3.371022333	1.073029735
8	⁠ $\scriptstyle 2+2{\sqrt {2}}$ ⁠	4.828427125	⁠ $\scriptstyle 2{\sqrt {2}}$ ⁠	2.828427125	0.9003163160	⁠ $\scriptstyle 8\left({\sqrt {2}}-1\right)$ ⁠	3.313708500	1.054786175
9		6.181824194		2.892544244	0.9207254290		3.275732109	1.042697914
10	⁠ $\scriptstyle {\tfrac {5}{2}}{\sqrt {5+2{\sqrt {5}}}}$ ⁠	7.694208843	⁠ $\scriptstyle {\tfrac {5}{2}}{\sqrt {{\tfrac {1}{2}}\left(5-{\sqrt {5}}\right)}}$ ⁠	2.938926262	0.9354892840	⁠ $\scriptstyle 2{\sqrt {25-10{\sqrt {5}}}}$ ⁠	3.249196963	1.034251515
11		9.365639907		2.973524496	0.9465022440		3.229891423	1.028106371
12	⁠ $\scriptstyle 6+3{\sqrt {3}}$ ⁠	11.19615242	3	3.000000000	0.9549296586	⁠ $\scriptstyle 12\left(2-{\sqrt {3}}\right)$ ⁠	3.215390309	1.023490523
13		13.18576833		3.020700617	0.9615188694		3.204212220	1.019932427
14		15.33450194		3.037186175	0.9667663859		3.195408642	1.017130161
15	⁠ $\scriptstyle {\tfrac {15}{8}}\left({\sqrt {15}}+{\sqrt {3}}+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\right)$ ⁠	17.64236291	⁠ $\scriptstyle {\tfrac {15}{16}}\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)$ ⁠	3.050524822	0.9710122088	⁠ $\scriptstyle {\tfrac {15}{2}}\left(3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {2\left(25-11{\sqrt {5}}\right)}}\right)$ ⁠	3.188348426	1.014882824
16	⁠ $\scriptstyle 4\left(1+{\sqrt {2}}+{\sqrt {2\left(2+{\sqrt {2}}\right)}}\right)$ ⁠	20.10935797	⁠ $\scriptstyle 4{\sqrt {2-{\sqrt {2}}}}$ ⁠	3.061467460	0.9744953584	⁠ $\scriptstyle 16\left(1+{\sqrt {2}}\right)\left({\sqrt {2\left(2-{\sqrt {2}}\right)}}-1\right)$ ⁠	3.182597878	1.013052368
17		22.73549190		3.070554163	0.9773877456		3.177850752	1.011541311
18		25.52076819		3.078181290	0.9798155361		3.173885653	1.010279181
19		28.46518943		3.084644958	0.9818729854		3.170539238	1.009213984
20	⁠ $\scriptstyle 5\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)$ ⁠	31.56875757	⁠ $\scriptstyle {\tfrac {5}{2}}\left({\sqrt {5}}-1\right)$ ⁠	3.090169944	0.9836316430	⁠ $\scriptstyle 20\left(1+{\sqrt {5}}-{\sqrt {5+2{\sqrt {5}}}}\right)$ ⁠	3.167688806	1.008306663
100		795.5128988		3.139525977	0.9993421565		3.142626605	1.000329117
1000		79577.20975		3.141571983	0.9999934200		3.141602989	1.000003290
10⁴		7957746.893		3.141592448	0.9999999345		3.141592757	1.000000033
10⁶		79577471545		3.141592654	1.000000000		3.141592654	1.000000000

Comparison of sizes of regular polygons with the same edge length, fromthree tosixty sides. The size increases without bound as the number of sides approaches infinity.

Of alln-gons with a given perimeter, the one with the largest area is regular.^[10]

Constructible polygon

[edit]

Main article:Constructible polygon

Some regular polygons are easy toconstruct with compass and straightedge; other regular polygons are not constructible at all.Theancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,^[11]^{: p. xi} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.^[11]^{: pp. 49–50} This led to the question being posed: is it possible to constructall regularn-gons with compass and straightedge? If not, whichn-gons are constructible and which are not?

Carl Friedrich Gauss proved the constructibility of the regular17-gon in 1796. Five years later, he developed the theory ofGaussian periods in hisDisquisitiones Arithmeticae. This theory allowed him to formulate asufficient condition for the constructibility of regular polygons:

A regularn-gon can be constructed with compass and straightedge ifn is the product of a power of 2 and any number of distinctFermat primes (including none).

(A Fermat prime is aprime number of the form $2^{\left(2^{n}\right)}+1.$ ) Gauss stated without proof that this condition was alsonecessary, but never published his proof. A full proof of necessity was given byPierre Wantzel in 1837. The result is known as theGauss–Wantzel theorem.

Equivalently, a regularn-gon is constructible if and only if thecosine of its common angle is aconstructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.

Regular skew polygons

[edit]

Thecube contains a skew regularhexagon, seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis.

The zig-zagging side edges of an-antiprism represent a regular skew 2n-gon, as shown in this 17-gonal antiprism.

Aregularskew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniformantiprism. All edges and internal angles are equal.

ThePlatonic solids (thetetrahedron,cube,octahedron,dodecahedron, andicosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively.

More generallyregular skew polygons can be defined inn-space. Examples include thePetrie polygons, polygonal paths of edges that divide aregular polytope into two halves, and seen as a regular polygon in orthogonal projection.

In the infinite limitregular skew polygons become skewapeirogons.

Regular star polygons

[edit]

Regular star polygons

2 < 2q < p,gcd(p, q) = 1

{p/q}

Vertices andEdges

Density

Coxeter diagram

Symmetry group

Dihedral (D_p)

Dual polygon

Self-dual

Internal angle
(degrees)

180-{\frac {360q}{p}}

^[12]

A non-convex regular polygon is a regularstar polygon. The most common example is thepentagram, which has the same vertices as apentagon, but connects alternating vertices.

For ann-sided star polygon, theSchläfli symbol is modified to indicate thedensity or "starriness"m of the polygon, as {n/m}. Ifm is 2, for example, then every second point is joined. Ifm is 3, then every third point is joined. The boundary of the polygon winds around the centerm times.

The (non-degenerate) regular stars of up to 12 sides are:

Pentagram – {5/2}
Heptagram – {7/2} and {7/3}
Octagram – {8/3}
Enneagram – {9/2} and {9/4}
Decagram – {10/3}
Hendecagram – {11/2}, {11/3}, {11/4} and {11/5}
Dodecagram – {12/5}

m andn must becoprime, or the figure will degenerate.

The degenerate regular stars of up to 12 sides are:

Tetragon – {4/2}
Hexagons – {6/2}, {6/3}
Octagons – {8/2}, {8/4}
Enneagon – {9/3}
Decagons – {10/2}, {10/4}, and {10/5}
Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6}

Two interpretations of {6/2}
Grünbaum {6/2} or 2{3}^[13]	Coxeter 2{3} or {6}[2{3}]{6}

Doubly-wound hexagon	Hexagram as a compound of two triangles

Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example, {6/2} may be treated in either of two ways:

For much of the 20th century (see for exampleCoxeter (1948)), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbors two steps away, to obtain the regularcompound of two triangles, orhexagram.
Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so thehexagram is represented as {6}[2{3}]{6}.^[14] More compactly Coxeter also writes 2{n/2}, like 2{3} for a hexagram as compound asalternations of regular even-sided polygons, with italics on the leading factor to differentiate it from the coinciding interpretation.^[15]
Many modern geometers, such as Grünbaum (2003),^[13] regard this as incorrect. They take the /2 to indicate moving two places around the {6} at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories ofabstract polytopes, but it also more closely copies the way in which Poinsot (1809) created his star polygons – by taking a single length of wire and bending it at successive points through the same angle until the figure closed.

Duality of regular polygons

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This sectionneeds expansion. You can help byadding to it.(December 2024)

Regular polygons as faces of polyhedra

[edit]

Auniform polyhedron has regular polygons as faces, such that for every two vertices there is anisometry mapping one into the other (just as there is for a regular polygon).

Aquasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex.

Aregular polyhedron is a uniform polyhedron which has just one kind of face.

The remaining (non-uniform)convex polyhedra with regular faces are known as theJohnson solids.

A polyhedron having regular triangles as faces is called adeltahedron.

Notes

[edit]

^(sequenceA007678 in theOEIS)

^Results forR = 1 anda = 1 obtained withMaple, using function definition:

f:=proc(n)optionsoperator,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)]]endproc

The expressions forn = 16 are obtained by twice applying thetangent half-angle formula to tan(π/4)

References

[edit]

^Hwa, Young Lee (2017).Origami-Constructible Numbers(PDF) (MA thesis). University of Georgia. pp. 55–59.
^Park, Poo-Sung. "Regular polytope distances",Forum Geometricorum 16, 2016, 227-232.http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf Archived 2016-10-10 at theWayback Machine
^^a ^b ^cMeskhishvili, Mamuka (2020)."Cyclic Averages of Regular Polygons and Platonic Solids".Communications in Mathematics and Applications.11:335–355.arXiv:2010.12340.doi:10.26713/cma.v11i3.1420 (inactive 1 July 2025).{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link)
^^a ^b ^c ^dJohnson, Roger A.,Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
^Pickover, Clifford A,The Math Book, Sterling, 2009: p. 150
^Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem",The College Mathematics Journal 37(5), 2006, pp. 390–391.
^Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^"Math Open Reference". Retrieved4 Feb 2014.
^"Mathwords".
^Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 inMathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
^^a ^bBold, Benjamin.Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).
^Kappraff, Jay (2002).Beyond measure: a guided tour through nature, myth, and number. World Scientific. p. 258.ISBN 978-981-02-4702-7.
^^a ^bAre Your Polyhedra the Same as My Polyhedra?Branko Grünbaum (2003), Fig. 3
^Regular polytopes, p.95
^Coxeter, The Densities of the Regular Polytopes II, 1932, p.53

External links

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Weisstein, Eric W."Regular polygon".MathWorld.
Regular Polygon description With interactive animation
Incircle of a Regular Polygon With interactive animation
Area of a Regular Polygon Three different formulae, with interactive animation
Renaissance artists' constructions of regular polygons Archived 2006-02-12 at theWayback Machine atConvergence Archived 2006-02-12 at theWayback Machine

Polygons (List)

Triangles

Quadrilaterals

By number
of sides

1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)

Star polygons

Classes

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

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