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.

Regular polygon

Edges and vertices	${\displaystyle n}$
Schläfli symbol	${\displaystyle \{n\}}$
Coxeter–Dynkin diagram
Symmetry group	Dn, order 2n
Dual polygon	Self-dual
Area (with side length ${\displaystyle s}$)	${\displaystyle A={\tfrac {1}{4}}ns^{2}\cot \left({\frac {\pi }{n}}\right)}$
Internal angle	${\displaystyle (n-2)\times {\frac {\pi }{n}}}$
Internal angle sum	${\displaystyle \left(n-2\right)\times {\pi }}$
Inscribed circle diameter	${\displaystyle d_{\text{IC}}=s\cot \left({\frac {\pi }{n}}\right)}$
Circumscribed circle diameter	${\displaystyle d_{\text{OC}}=s\csc \left({\frac {\pi }{n}}\right)}$
Properties	Convex, cyclic, equilateral, isogonal, isotoxal