Areas of polygons inscribed in a circle

Abstract

Heron of Alexandria showed that the area K of a triangle with sides a, b, and c is given by {Mathematical expression} where s is the semiperimeter (a+b+c)/2. Brahmagupta gave a generalization to quadrilaterals inscribed in a circle. In this paper we derive formulas giving the areas of a pentagon or hexagon inscribed in a circle in terms of their side lengths. While the pentagon and hexagon formulas are complicated, we show that each can be written in a surprisingly compact form related to the formula for the discriminant of a cubic polynomial in one variable. © 1994 Springer-Verlag New York Inc.