Proof 1. Let stand for the volume of a solid . Let be the edge length of the large tetrahedron . Then a regular tetrahedron with edge length has volume for some . We get a regular octahedron by cutting away four regular tetrahedra from the large tetrahedron. So .

Proof 2. Let a skew prism with equilateral triangular base be decomposed into a regular tetrahedron and into a square pyramid having all edges of the same length. This pyramid is half of a regular octahedron. But the volume of the tetrahedron is one-third of the volume of the prism, and the volume of the pyramid is two-thirds of the volume of the prism. So the volume of the octahedron is four times the volume of the tetrahedron.