The regular tetrahedron has the same volume as a quarter octahedron, which is as well a tetrahedron, whose four faces are two equilateral triangles and the other two are isosceles right triangles (half squares) that intersect at right angles.[br][br]We show that these two tetrahedra have the same volume using Cavalieri's principle: by choosing an equilateral triangle as the base, the apex can move in a horizontal plane without changing the volume of the corresponding tetrahedron. And we go from one to another in symmetry with respect to a vertical plane that contains a base edge.

Prove this volume equality analytically. What is the volume of a tetrahedron? Of a more general pyramid?