Tetrahedron in Cube

A tetrahedron inscribed in a cube. This is a study of the volume of the tetrahedron relative to the cube.

A 3D design created in vZome. Use your mouse or touch to interact.

This shows the cube with copies of the 4 isosceles tetrahedra that fill out the space in the cube that is not occupied by the regular tetrahedron pulled out.

This shows a double edge length tetrahedron, with an octahedron inscribed. Given that a double edge length solid has a volume that is 8 times the volume of a single edge length of the same solid, this has a volume of 8 of the small tetrahedra. Copies of the 4 isosceles tetrahedra taken from the dissection of the cube are shown as pulled out from, and in place as half of the volume of the octahedron. Subtracting 4 regular tetrahedra from the total of 8, this says that the volume of the octahedron is 4 tetrahedron units, and the 4 isosceles tetrahedra have a combined volume of 2 units, so the regular tetrahedron occupies one third of the volume of the cube.