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

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About mvtrinh

Retired high school math teacher.
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