Regular polygon

Set of regular p-gons
Regular polygons
Edges and vertices	p
Schläfli symbol	{p}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (Dp)
Dual polyhedron	Self-dual
Area (with t=edge length)	$A={\frac {pt^{2}}{4\tan(\pi /p)}}$
Internal angle (degrees)	$\left(1-{\frac {2}{p}}\right)\times 180$

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

General properties

These properties apply to both convex and star regular polygons.

All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points, i.e., every regular polygon has a circumscribed circle.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle.

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 (there is rotational symmetry of order 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 which 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: degenerate in ordinary space {1}
• Digon: a "double line segment" - degenerate in ordinary space {2}
• Equilateral triangle {3}
• Square {4}
• Regular pentagon {5}
• Regular hexagon {6}
• Regular heptagon {7}
• Regular octagon {8}
• Regular enneagon or nonagon {9}
• Regular decagon {10}
• Regular hendecagon {11}
• Regular dodecagon {12}

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.

Properties

Each interior angle of a regular convex n-gon 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.

Each supplementary angle has a measure of ${\frac {360}{n}}$ degrees, and the sum of the supplementary angles is 360 degrees or 2π radians or one full turn.

For $n>2$ the number of diagonals is ${\frac {n(n-3)}{2}}$, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.

Area

Apothem of a hexagon

The area A of a convex regular n-sided polygon is:

in degrees

$A={\frac {nt^{2}}{4\tan({\frac {180}{n}})}}$,

or in radians

$A={\frac {nt^{2}}{4\tan({\frac {\pi }{n}})}}$,

where t is the length of a side.

If the radius, or length of the segment joining the center to the vertex is known, the area is:

in degrees

$A={\frac {nr^{2}sin({\frac {360}{n}})}{2}}$

or in radians

$A={\frac {nr^{2}sin({\frac {2\pi }{n}})}{2}}$,

where r is the radius

Also, the area is half the perimeter multiplied by the length of the apothem, a, (the line drawn from the centre of the polygon perpendicular to a side). That is A = a.n.t/2, as the length of the perimeter is n.t, or more simply 1/2 p.a.

For sides t=1 this gives:

in degrees

${\frac {n}{4\tan({\frac {180}{n}})}}$

or in radians (n not equal to 2)

${\frac {n}{4}}\cot(\pi /n)$

with the following values:

Sides	Name	Exact area	Approximate area
3	equilateral triangle	${\frac {\sqrt {3}}{4}}$	0.433
4	square	1	1.000
5	regular-pentagon	${\frac {1}{4}}{\sqrt {25+10{\sqrt {5}}}}$	1.720
6	regular-hexagon	${\frac {3{\sqrt {3}}}{2}}$	2.598
7	regular-heptagon		3.634
8	regular-octagon	$2+2{\sqrt {2}}$	4.828
9	regular-enneagon		6.182
10	regular-decagon	${\frac {5}{2}}{\sqrt {5+2{\sqrt {5}}}}$	7.694
11	regular-hendecagon		9.366
12	regular-dodecagon	$6+3{\sqrt {3}}$	11.196
13	regular-triskaidecagon		13.186
14	regular-tetradecagon		15.335
15	regular-pentadecagon		17.642
16	regular-hexadecagon		20.109
17	regular-heptadecagon		22.735
18	regular-octadecagon		25.521
19	regular-enneadecagon		28.465
20	regular-icosagon		31.569
100	regular-hectagon		795.513
1000	regular-chiliagon		79577.210
10000	regular-myriagon		7957746.893

The amounts that the areas are less than those of circles with the same perimeter, are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing n to the limit π/12).

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 '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 centre m times, and m is sometimes called the density of the polygon.

Examples:

• Pentagram - {5/2}
• Heptagram - {7/2} and {7/3}
• Octagram - {8/3}
• Enneagram - {9/2} and {9/4}
• Decagram - {10/3}

m and n must be co-prime, or the figure will degenerate. 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 neighbours 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.

Regular polygons as faces of polyhedra

A uniform polyhedron is a polyhedron with 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).

The remaining convex polyhedra with regular faces are known as the Johnson solids.

References

• Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.
• 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.

See also

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

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