Regular polygon

Definition

Let $n \geq 3$, and let $P_1 P_2 \cdots P_n$ be a convex polygon in the Euclidean plane $\mathbb{R}^2$. 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 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, 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.