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 or star. 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.

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

n

=

2

a

3

b

p

1

⋯

p

r

{\displaystyle n=2^{a}3^{b}p_{1}\cdots p_{r}}

for some

r

∈

N

{\displaystyle r\in \mathbb {N} }

, where each distinct

p

i

{\displaystyle p_{i}}

is a Pierpont prime.