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

**SOLUTION**

**
**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

Simplify

**Answer**: