Discrete & Computational Geometry

, Volume 12, Issue 2, pp 223-236

First online:

Areas of polygons inscribed in a circle

  • D. P. RobbinsAffiliated withCenter for Communications Research, Institute for Defense Analyses

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Heron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by
$$K = \sqrt {s(s - a)(s - b)(s - c)} ,$$
wheres 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.