Schlafli's volume formula and geometrization
The volume of a Euclidean polyhedron cannot be determined from its dihedral angles, because we can always scale the polyhedron up or down. This is no longer true in spherical or hyperbolic geometry, where the space has uniform sectional curvature. After a quick historical overview, I will sketch a proof of Schlafli's differential formula, which relates the variations of the volume and of the dihedral angles. Since geometers care more about manifolds than about polyhedra, I will explain an important application due to Rivin: if a manifold is decomposed into polyhedra, finding a
complete hyperbolic (or spherical) metric on the manifold is equivalent to finding a critical point for the sum of the
volumes of the polyhedra.