This is mostly about tetrahedra: how Eudoxus cleverly proved the formula for
their volume, then the volume of a cone, how Archimedes then proved the
formula for the volume of a sphere, then (fast forward two thousand years)
how Legendre used the Herriot's formula for the area of a spherical triangle
to prove Euler's formula F - E + V = 2 for convex polyhedra, then more about
tetrahedra: simple vector proofs of theorems of Monge (1810) and Mannheim
(1895) on the Monge point of a tetrahedron (somewhat analogous to the
orthocenter of a triangle.