The perimeter and volume of a Reuleaux polyhedron

Abstract

A ball polyhedron is the intersection of a finite number of closed balls in $\mathbb{R}^3$ with the same radius. In this note, we study ball polyhedra in which the set of centers defining the balls have the maximum possible number of diametric pairs. We explain how to compute the perimeter and volume of these shapes by employing the Gauss-Bonnet theorem and another integral formula. In addition, we show how to adapt this method to approximate the volume of Meissner polyhedra, which are constant width bodies constructed from ball polyhedra.