2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68535We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class-preserving symmetries of the space of (suitably decorated) generalized hyperbolic tetrahedra. The group 22.5K contains the Regge symmetries as a subgroup of order 144. From a generic tetrahedron, 22.5K produces 30 distinct generalized tetrahedra in the same scissors class, including the 12 honest-to-goodness tetrahedra produced by the Regge subgroup. The action of 22.5K leads us to the Murakami-Yano formula, and to 9 others, which are similar but less symmetrical. From here, we can derive yet other volume formulas with pleasant algebraic and analytical properties. The key to understanding all this is a natural relationship between a hyperbolic tetrahedron and a pair of ideal hyperbolic octahedra.50 pages with 12 figuresGeometric TopologyMetric Geometry51M10 (Primary), 51M20, 57M99 (Secondary)text