Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.

1 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. That is, a regular polygon is a cyclic polygon.

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 midpoint. Thus a regular polygon is a tangential polygon.

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.

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

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

1.1 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, 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.

2 Regular convex polygons

2.1 Angles

2.2 Diagonals

2.3 Points in the plane

2.3.1 Interior points

2.4 Circumradius

2.5 Dissections

2.6 Area

3 Constructible polygon

4 Regular skew polygons

5 Regular star polygons

6 Duality of regular polygons

7 Regular polygons as faces of polyhedra

8 See also