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5.3 Regular Polygons

  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=½ka 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

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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.