Regular polygon

A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length).

All regular polygons with the same number of sides are similar.

Regular digon: degenerate, a "double line segment"
Equilateral triangle (3 side)
Square (4 sides)
Regular pentagon (5 sides)
Regular hexagon (6 sides)
Regular heptagon (7 sides)
Regular octagon (8 sides)
Regular decagon (10 sides)
Regular dodecagon (12 sides)

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.

Properties

Each angle of a regular n-gon has a measure of $(1-{\frac {2}{n}})\times 180$ (or equally of $(n-2)\times {\frac {180}{n}}$ ) degrees.

Alternately, the internal angle(s) of a regular n-gon is ${\frac {(n-2)\pi }{n}}$ radians ( or ${\frac {(n-2)}{2n}}$ turns).

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.

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.

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

The area of a regular n-sided polygon is

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

where t is the length of a side. Also, the area is half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side).

For t=1 this gives

{\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).

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.

Nonconvex regular polygons

An extended category of regular polygons includes the star polygons, for example a pentagram, which has the same vertices as a pentagon, but connects alternating vertices.

Examples:

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

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.

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