## Volume of Octahedron

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 $2a$. The octahedron is made up of an upper pyramid and a lower pyramid. The height $h$ of each pyramid equals $a$. The base of the pyramids is a square of side length $a\sqrt 2$. The surface area $B$ of the base equals $2a^2$.
Volume of octahedron = volume of upper pyramid + volume of lower pyramid
$=Bh/3+Bh/3$
$=2Bh/3$
$=2(2a^2)(a)/3$
$=4a^3/3$
Volume of cube = $(2a)^3$
$=8a^3$
Ratio of volume of cube to volume of octahedron
$8a^3 : 4a^3/3$
Simplify
$6 : 1$

Answer: $6 : 1$