Integrated Circumradius and Area Formulae for Cyclic Pentagons and Hexagons

Abstract

This paper describes computations of the relations between the circumradius R and area S of cyclic polygons given by the lengths of the sides. The classic results of Heron and Brahmagupta clearly show that the product of R and S is expressed by the lengths of the sides for triangles and cyclic quadrilaterals. However, the formulae of circumradius and area for cyclic pentagons and hexagons have been studied separately, and the relation between them has seldom been discussed. In this study, based on the results derived by Robbins 1994, Pech 2006, and the author 2011, we succeeded in computing integrated formulae of the circumradius and the area for cyclic pentagons and hexagons. They are found to be a polynomial equation in $$4SR^2$$4SR2 with degree 7 for pentagons, and the product of two polynomials each with degree 7 for hexagons. We confirmed that these three polynomials with degree 7 are uniformly expressed using the notion of crossing parity, the structure of which is analogous to those of the area formulae and circumradius formulae for $$n=5,6$$n=5,6. Moreover, we derived a polynomial equation in 4SR itself with degree 7 for cyclic pentagons, and showed that this type of formula exists only for n-gons, where n is an odd number.