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

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

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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.

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Contributed by: Izidor Hafner (November 2014)
Open content licensed under CC BY-NC-SA


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