Two Proofs that the Volume of the Regular Octahedron Is Four Times the Volume of the Regular Tetrahedron

This Demonstration shows two visual proofs that the volume of the regular octahedron is four times that of the regular tetrahedron.

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.