  A ball polyhedron is the intersection of a finite number of closed balls 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.
