  A three-dimensional orthoscheme is defined as a tetrahedron whose base is a
right-angled triangle and an edge joining the apex and a non-right-angled
vertex is perpendicular to the base. A generalization, called complete
orthoschemes, of orthoschemes is known in hyperbolic geometry. Roughly
speaking, complete orthoschemes consist of three kinds of polyhedra; either
compact, ideal or truncated. We consider a particular family of hyperbolic
complete orthoschemes, which share the same base. They are parametrized by the
"height", which represents how far the apex is from the base. We prove that the
volume attains maximal when the apex is ultraideal in the sense of hyperbolic
geometry, and that such a complete orthoscheme is unique in the family.
