# Definition:Polygon/Regular

## Definition

A regular polygon is a polygon which is both equilateral and equiangular.

That is, in which all the sides are the same length, and all the vertices have the same angle: ### Center

The center of a regular polygon $P$ is defined as the point which is the center of the circumcircle of $P$. In the above, $O$ is the center of the regular polygon.

The long radius of a regular polygon $P$ is defined as the distance from the center of $P$ to one of its vertices. In the above, the length of $OA$ is the long radius of the regular polygon.

### Apothem

The apothem of a regular polygon $P$ is defined as the perpendicular distance from the center of $P$ to one of its sides. In the above, the length of $OM$ is the apothem of the regular polygon.

## Also known as

In Euclid's The Elements, a regular polygon is referred to as an equilateral and equiangular polygon.

Some sources use the word perfect or symmetrical instead of regular.

## Examples

Specific instances of regular polygons with specific numbers of sides are as follows:

The term regular $n$-gon is usually used nowadays to specify a regular polygon with a specific number, that is $n$, sides.

The specific name is usually invoked only in order to draw attention to the fact that such a regular polygon has a particularly interesting set of properties.

## Also see

• Results about regular polygons can be found here.