An octahedron is formed by connecting the centers of the faces of a cube. What is the ratio of the volume of the cube to that of the contained octahedron?
Source: NCTM Mathematics Teacher 2006
Suppose the cube side length equals . The octahedron is made up of an upper pyramid and a lower pyramid. The height of each pyramid equals . The base of the pyramids is a square of side length . The surface area of the base equals .
Volume of octahedron = volume of upper pyramid + volume of lower pyramid
Volume of cube =
Ratio of volume of cube to volume of octahedron