Areas of polygons inscribed in a circle

AbstractHeron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9vqFf0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVe0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2% da9maakaaabaGaam4CaiaacIcacaWGZbGaeyOeI0IaamyyaiaacMca% caGGOaGaam4CaiabgkHiTiaadkgacaGGPaGaaiikaiaadohacqGHsi% slcaWGJbGaaiykaaWcbeaakiaacYcaaaa!4935! $$K = \sqrt {s(s - a)(s - b)(s - c)} ,$$ 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.