This tetrahedron has two faces which are equilateral triangles and the other two are isosceles right triangles (half squares) that intersect at right angles. It forms a quarter octahedron of the same volume as the regular tetrahedron with which it shares an equilateral triangle.[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 vertex 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?