Two tetrahedra of the same volume
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.
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?