What Fits in a Cube

Each solid fits perfectly in the glass cube. Can you fit the red tetrahedron in the cube? Hint: How many edges does the tetrahedron have?

Can you fit the blue Kepler Star and the yellow cubeoctahedron in the cube? Hint: Examine features such as edges, faces and corners and relate those to the cube.

What percentage of the cube’s volume is occupied by each solid?

Notice the tetrahedron has six edges and that a cube has six faces. This is not just a coincidence, but a geometric relationship between the tetrahedron and the cube. When you slide the tetrahedron into the cube there will be exactly one edge on each face of the cube. When you slide the tetrahedron into the cube, the four corners of the tetrahedron occupies four corners of the cube – leaving the other four corners free. The Kepler Star is made by intersecting two tetrahedrons, so it has eight corners that can occupy all the corners of the cube. Finally, the cubeoctahedron is made from eight triangle faces and six square faces. When you slide the solid into the glass cube, each square face will meet each face of the cube. In fact, the cubeoctahedron is made by snipping off the corners of a cube.