# Regular polygon

## Equiangular and equilateral polygon / From Wikipedia, the free encyclopedia

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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.

Equiangular and equilateral polygon

**Quick facts: , Edges and vertices, Schläfli symbol, Coxete...**▼

Edges and vertices | $n$ | ||||||||||||||||||||
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Schläfli symbol | $\{n\}$ | ||||||||||||||||||||

Coxeter–Dynkin diagram | |||||||||||||||||||||

Symmetry group | D_{n}, order 2n | ||||||||||||||||||||

Dual polygon | Self-dual | ||||||||||||||||||||

Area (with side length $s$) | $A={\tfrac {1}{4}}ns^{2}\cot \left({\frac {\pi }{n}}\right)$ | ||||||||||||||||||||

Internal angle | $(n-2)\times {\frac {\pi }{n}}$ | ||||||||||||||||||||

Internal angle sum | $\left(n-2\right)\times {\pi }$ | ||||||||||||||||||||

Inscribed circle diameter | $d_{\text{IC}}=s\cot \left({\frac {\pi }{n}}\right)$ | ||||||||||||||||||||

Circumscribed circle diameter | $d_{\text{OC}}=s\csc \left({\frac {\pi }{n}}\right)$ | ||||||||||||||||||||

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

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