A trick for calculating volumes of non-Euclidean polyhedra
Using the proof idea for a formula for the volume of a compact tetrahedron in three-dimensional hyperbolic space \({\mathbb H}^3\) proposed by \textit{G. Sforza} [Modena Mem. (3) 8, 21--66 (1907; JFM 38.0675.01)], the author establishes alternative formulas for the volumes of hyperbolic orthoschemes, of hyperbolic octahedra with \(4|m\)-symmetry (invariant under the rotation about the axis joining the pyramid vertices through an angle of \(\frac{\pi}{2}\) and reflection with respect to the plane perpendicular to it) whose dihedral angles satisfy a certain system of inequalities, and of spherical octahedra with \(4|m\)-symmetry.
scientific article; zbMATH DE number 7369743