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A polygon is **regular** if all its sides are equal and all its
angles are equal. Either condition implies the other in the case of a
triangle, but not in general. (A rhombus has equal sides but not
necessarily equal angles, and a rectangle has equal angles but not
necessarily equal sides.)

For a *k*-sided regular polygon of side *a*, let be the angle
at any vertex, *r* and *R* the radii of the inscribed and
circumscribed circles (*r* is called the **apothem**). As usual, let
*s*=½*ka* be the half-perimeter. Then:

If denotes the side of a *k*-sided regular polygon
inscribed in a circle of radius *R*, we have

If denotes the side of a *k*-sided regular polygon
circumscribed about the same circle, we have

In particular,

The areas , , and of the same polygons satisfy

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.