---
tag: 0V63D9V0FY
kind: definition
name: Regular polygon
msc: [51M20]
defines: 01KJJ9A400
uses: [0XW7V9ZZ05, 0NPPGHQNXQ, 0GJAJC6TNZ, 049DCJZ448]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_polygon
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Regular_polygon
    locator: General properties
  - url: https://en.wikipedia.org/wiki/Regular_polygon
    locator: Angles
---

## Definition

Let $n \geq 3$, and let $P_1 P_2 \cdots P_n$ be a convex polygon in the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Write $P_0 = P_n$ and $P_{n+1} = P_1$.
The polygon is regular if all sides have the same length, $d(P_i, P_{i+1}) = d(P_1, P_2)$, and all [[angles|0GJAJC6TNZ]] are equal, $\angle P_{i-1} P_i P_{i+1} = \angle P_n P_1 P_2$, for $i = 1, \dots, n$.

## Remarks

Each interior angle of a regular convex $n$-gon is $\frac{(n - 2)\pi}{n}$. Each exterior angle is $\frac{2\pi}{n}$.
The vertices are [[concyclic|049DCJZ448]], and the polygon also has an incircle. The vertices of a regular $n$-gon with the circumradius $1$ can be $\left(\cos \frac{2\pi k}{n}, \sin \frac{2\pi k}{n}\right)$, for $k = 0, \dots, n - 1$.
With the side $s$, the area is $\frac{1}{4} n s^2 \cot \frac{\pi}{n}$.
Star polygons, such as the pentagram, are also called regular polygons. They are not convex.

## Explanations

### L1 human

A regular polygon looks the same from each corner. Equal sides alone are not enough: a rhombus has four equal sides but is regular only when it is a square. Equal angles alone are not enough either, as a rectangle shows.