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 or star. 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.

These properties apply to all regular polygons, whether convex or star:

The symmetry group of an n-sided regular polygon is the dihedral group D_n (of order 2n): D₂, D₃, D₄, ... It consists of the rotations in C_n, together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side.

All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.

An n-sided convex regular polygon is denoted by its Schläfli symbol ${\displaystyle \{n\}}$. For ${\displaystyle n<3}$, we have two degenerate cases:

In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc.

For a regular convex n-gon, each interior angle has a measure of:

and each exterior angle (i.e., supplementary to the interior angle) has a measure of ${\displaystyle {\tfrac {360}{n}}}$ degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.