The Volume of the Regular Octahedron Is Four Times the Volume of the Regular Tetrahedron

This Demonstration shows a visual proof that the volume of the regular octahedron is four times that of the regular tetrahedron through decomposition. The large octahedron has a side that is twice the length of any of the small octahedra. So the volume of the large octahedron is eight times as much as a small one. But the large octahedron is made of six small octahedra and eight tetrahedra. So the eight tetrahedra must have a volume equal to two small octahedra, and the ratio is 4 to 1.

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