the regular polygons

The regular polygons are the analogues, in dimension two, of the regular polyhedra in dimension three and of the regular polytopes in dimension four. A regular polygon has all its sides equal and all its angles equal; its vertices are regularly distributed on a circle (their number n>2 is the order of the polygon).
The number of regular polygons is infinite, while there exists only a finite number of regular polyhedra (nine) and of regular polytopes (sixteen). All these polygons are not constructible with compass and straightedge.
We distinguish the convex polygons (their n(n-3)/2 diagonals are inside) and the star polygons (their sides are n diagonals of the convex ones). The star figures of order n are compounds of identical regular polygons of order k; they exist only for non prime orders greater than 5 (k>2 divides n).
The faces of the uniform polyhedra, therefore of the regular polyhedra and of the Archimedes' polyhedra, are regular polygons.

Here are all the regular polygons of orders 3 to 14; the star polygons and the star figures are shown under the convex polygons of same order.

Remark: the areas of a square and a convex regular dodecagon inscribed in a circle of radius r are respectively 2r² and 3r².