Integrating Circumradius and Area Formulae for Cyclic Pentagons

Abstract

This paper describes computations of the relations between circumradius R and area S of cyclic polygons given by the lengths of the sides. Classic results by Heron and Brahmagupta clearly show the relation of circumradius and area for triangles and cyclic quadrilaterals. In contrast, formulae for the circumradius and the area of cyclic pentagons have been studied separately. D.P.Robbins obtained the area formula in 1994, which is a polynomial equation in (4S)² with degree 7. The circumradius formula was given by P.Pech in 2006, which is also a polynomial equation in R ² with degree 7. In this study, we succeeded in computing the integrated formula for the circumradius and the area of cyclic pentagons. It is found to be a polynomial equation in (4SR) with degree 7. This equation is easily transformed into the equation in (4SR)² with degree 7, hence both the expressions are meaningful. The existence of the latter form of formula was pointed out by D.Svrtan et al. in 2004, but somehow their result seems to contain typographical errors. Therefore, we believe that our results correspond to the correction and extension of already known formulae.

This work was supported by a Grant-in-Aid for Scientific Research (25330006) from the Japan Society for the Promotion of Science (JSPS).