ON THE VOLUME OF A SPHERICAL OCTAHEDRON WITH SYMMETRIES

摘要

In the present paper, closed integral formulas for the volumes of spherical octahedra and hexahedra having nontrivial symmetries are established. Trigonometrical identities involving lengths of edges and dihedral angles (the sine-tangent rules) are obtained. This gives the possibility of expressing the lengths in terms of angles. Then the Schlaefli formula is applied to find the volume of polyhedra in terms of dihedral angles explicitly. These results and the canonical duality between octahedra and hexahedra in the spherical space allowed us to express the volume in terms of the lengths of edges as well.