Regular polygon

Set of convex regular p-gons
Regular polygons
Edges and vertices	n
Schläfli symbol	{n}
Coxeter–Dynkin diagram
Symmetry group	Dn, order 2n
Dual polygon	Self-dual
Area (with t=edge length)	$A={\tfrac {1}{4}}nt^{2}\cot {\frac {\pi }{n}}$
Internal angle	$\left(1-{\frac {2}{n}}\right)\times 180^{\circ }$
Internal angle sum	$\left(n-2\right)\times 180^{\circ }$
Properties	convex, cyclic, equilateral, isogonal, isotoxal

A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star.

General properties

These properties apply to all regular polygons, whether convex or star.

A regular n-sided polygon has rotational symmetry of order n.

All vertices of a regular polygon lie on a common circle (the circumscribed circle), i.e., they are concyclic points.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the mid-point.

A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

Symmetry

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

Regular convex polygons

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

An n-sided convex regular polygon is denoted by its Schläfli symbol {n}.

• Henagon or monogon {1}: degenerate in 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 polygon).
• Digon {2}: a "double line segment": degenerate in ordinary space (Some authorities do not regard the digon as a true polygon because of this).

Equilateral triangle {3}	Square {4}	Pentagon {5}	Hexagon {6}	Heptagon {7}	Octagon {8}	Enneagon {9}	Decagon {10}
Hendecagon {11}	Dodecagon {12}	Tridecagon {13}	Tetradecagon {14}	Pentadecagon {15}	Hexadecagon {16}	Heptadecagon {17}	Octadecagon {18}	Enneadecagon {19}
Icosagon {20}	Triacontagon {30}	Tetracontagon {40}	Pentacontagon {50}	Hexacontagon {60}	Heptacontagon {70}	Octacontagon {80}	Enneacontagon {90}	Hectogon {100}

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.

Angles

For a regular convex n-gon, each interior angle has a measure of:

$(1-{\frac {2}{n}})\times 180$ (or equally of $(n-2)\times {\frac {180}{n}}$ ) degrees,

or ${\frac {(n-2)\pi }{n}}$ radians,

or ${\frac {(n-2)}{2n}}$ full turns,

and each exterior angle (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.

Diagonals

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

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

Interior points

For a regular n-gon, the sum of the perpendicular distances from any interior point to the n sides is n times the apothem (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n=3 case.

Circumradius

The circumradius from the center of a regular polygon to one of the vertices is related to the side length, s or apothem, a:

$r={\frac {s}{2\sin {\frac {\pi }{n}}}}={\frac {a}{\cos {\frac {\pi }{n}}}}$

Area

Regular polygon with n = 5: pentagon with side s, circumradius r and apothem a

The area A of a convex regular n-sided polygon having side s, circumradius r, apothem a, and perimeter p is given by

$A={\tfrac {1}{2}}nsa={\tfrac {1}{2}}pa={\tfrac {1}{4}}ns^{2}\cot {\tfrac {\pi }{n}}=na^{2}\tan {\tfrac {\pi }{n}}={\tfrac {1}{2}}nr^{2}\sin {\tfrac {2\pi }{n}}$

For regular polygons with side s=1, resp. circumradius r=1, resp. apothem a=1, this produces the following table:

Number of sides	Name of polygon	Area when side s=1		Area when circumradius r=1			Area when apothem a=1
		Exact	Approximate	Exact	Approximate	Approximate as fraction of circle	Exact	Approximate	Approximate as fraction of circle
n	regular n-gon	${\tfrac {n}{4}}\cot {\tfrac {\pi }{n}}$		${\tfrac {n}{2}}\sin {\tfrac {2\pi }{n}}$		${\tfrac {n}{2\pi }}\sin {\tfrac {2\pi }{n}}$	$n\tan {\tfrac {\pi }{n}}$		${\tfrac {n}{\pi }}\tan {\tfrac {\pi }{n}}$
3	equilateral triangle	√3/4	0.433012702	3√3/4	1.299038105	0.4134966714	3√3	5.196152424	1.653986686
4	square	1	1.000000000	2	2.000000000	0.6366197722	4	4.000000000	1.273239544
5	regular pentagon	1/4√25+10√5	1.720477401	5/4√(5+√5)/2	2.377641291	0.7568267288	5√5-2√5	3.632712640	1.156328347
6	regular hexagon	3√3/2	2.598076211	3√3/2	2.598076211	0.8269933428	2√3	3.464101616	1.102657791
7	regular heptagon		3.633912444		2.736410189	0.8710264157		3.371022333	1.073029735
8	regular octagon	2+2√2	4.828427125	2√2	2.828427125	0.9003163160	8(√2-1)	3.313708500	1.054786175
9	regular nonagon		6.181824194		2.892544244	0.9207254290		3.275732109	1.042697914
10	regular decagon	5/2√5+2√5	7.694208843	5/2√(5-√5)/2	2.938926262	0.9354892840	2√25-10√5	3.249196963	1.034251515
11	regular hendecagon		9.365639907		2.973524496	0.9465022440		3.229891423	1.028106371
12	regular dodecagon	6+3√3	11.19615242	3	3.000000000	0.9549296586	12(2-√3)	3.215390309	1.023490523
13	regular triskaidecagon		13.18576833		3.020700617	0.9615188694		3.204212220	1.019932427
14	regular tetradecagon		15.33450194		3.037186175	0.9667663859		3.195408642	1.017130161
15	regular pentadecagon		17.64236291		3.050524822	0.9710122088		3.188348426	1.014882824
16	regular hexadecagon		20.10935797	4√2-√2	3.061467460	0.9744953584		3.182597878	1.013052368
17	regular heptadecagon		22.73549190		3.070554163	0.9773877456		3.177850752	1.011541311
18	regular octadecagon		25.52076819		3.078181290	0.9798155361		3.173885653	1.010279181
19	regular enneadecagon		28.46518943		3.084644958	0.9818729854		3.170539238	1.009213984
20	regular icosagon	5 (1+√5+√5+2√5)	31.56875757	5/2 (√5-1)	3.090169944	0.9836316430	20 (1+√5-√5+2√5)	3.167688806	1.008306663
100	regular hectagon		795.5128988		3.139525977	0.9993421565		3.142626605	1.000329117
1000	regular chiliagon		79577.20975		3.141571983	0.9999934200		3.141602989	1.000003290
10000	regular myriagon		7957746.893		3.141592448	0.9999999345		3.141592757	1.000000033
1,000,000	regular megagon		79,577,471,545.685		3.141592654	1.000000000		3.141592654	1.000000000

Of all n-gons with a given perimeter, the one with the largest area is regular.

Regular skew polygons

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

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

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

The Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively.

More generally regular skew polygons can be defined in n-space. Examples include the Petrie polygons, polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection.

In the infinite limit regular skew polygons become skew apeirogons.

Regular star polygons

A pentagram {5/2}

A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, but connects alternating vertices.

For an n-sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as {n/m}. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of the polygon winds around the center m 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 and n must be co-prime, or the figure will degenerate.

The degenerate regular stars of up to 12 sides are:

• Hexagram – {6/2}
• Octagram – {8/2}
• Enneagram – {9/3}
• Decagram – {10/2} and {10/4}
• Dodecagram – {12/2}, {12/3} and {12/4}

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 example Coxeter (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 regular compound of two triangles, or hexagram.
• Many modern geometers, such as Grünbaum (2003), 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 of abstract 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

See also: Self-dual polyhedra

All regular polygons are self-dual to congruency, and for odd n they are self-dual to identity.

In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.

Regular polygons as faces of polyhedra

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

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

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

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

A polyhedron having regular triangles as faces is called a deltahedron.

See also

• Tiling by regular polygons
• Platonic solids
• Apeirogon – An infinite-sided polygon can also be regular, {∞}.
• List of regular polytopes
• Equilateral polygon

References

• Coxeter, H.S.M. (1948). "Regular Polytopes" (Document). Methuen and Co. {{cite document}}: Invalid |ref=harv (help)
• Grünbaum, B.; Are your polyhedra the same as my polyhedra?, Discrete and comput. geom: the Goodman-Pollack festschrift, Ed. Aronov et al., Springer (2003), pp. 461–488.
• Poinsot, L.; Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9 (1810), pp. 16–48.

External links

• 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 at Convergence