Definitions

# 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)
$A=frac\left\{pt^2\right\}\left\{4tan\left(pi/p\right)\right\}$
Internal angle
(degrees)
$left\left(1-frac\left\{2\right\}\left\{p\right\}right\right)times 180$
Internal angle sum
(degrees)
$left\left(p-2right\right)times 180$
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 Dn (of order 2n): D2, D3, D4,... It consists of the rotations in Cn (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}.

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:

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

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

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

and each exterior angle (supplementary to the interior angle) has a measure of $frac\left\{360\right\}\left\{n\right\}$ 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 $frac\left\{n \left(n-3\right)\right\}\left\{2\right\}$, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.

### Area

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

in degrees

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

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

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

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

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

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

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

or in radians (n not equal to 2)

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

with the following values:

Sides Name Exact area Approximate area
3 equilateral triangle $frac\left\{sqrt\left\{3\right\}\right\}\left\{4\right\}$ 0.433
4 square 1 1.000
5 regular-pentagon $frac \left\{1\right\}\left\{4\right\} sqrt\left\{25+10sqrt\left\{5\right\}\right\}$ 1.720
6 regular-hexagon $frac\left\{3 sqrt\left\{3\right\}\right\}\left\{2\right\}$ 2.598
7 regular-heptagon   3.634
8 regular-octagon $2 + 2 sqrt\left\{2\right\}$ 4.828
9 regular-enneagon   6.182
10 regular-decagon $frac\left\{5\right\}\left\{2\right\} sqrt\left\{5+2sqrt\left\{5\right\}\right\}$ 7.694
11 regular-hendecagon   9.366
12 regular-dodecagon $6+3sqrt\left\{3\right\}$ 11.196
13 regular-triskaidecagon   13.186
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 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:

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 ), 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.

## Self-dual polygons

All regular polygons are self-dual, with equal number of vertices and edges.

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

## 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

• 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.