Regular polygon

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

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.