# Regular polygon

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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.

Quick facts: , Edges and vertices, Schläfli symbol, Coxete...
Regular polygon
Edges and vertices${\displaystyle n}$
Schläfli symbol${\displaystyle \{n\}}$
Coxeter–Dynkin diagram
Symmetry groupDn, order 2n
Dual polygonSelf-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${\displaystyle d_{\text{OC}}=s\csc \left({\frac {\pi }{n}}\right)}$
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
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