2nd Mexico 1988 Problem B4 Calculate the volume of an octahedron which has an inscribed sphere of radius 1.
4√3 Solution Opposite faces of the octahedron are parallel and hence a distance 2 apart. Suppose the edge length is x. Then the distance from the center of a face to a vertex of the face is x/√3. A section through the centers of two opposite faces, perpendicular to the faces, and through a vertex of one, is shown above. Thus 4 = 3x 2 /4 - x 2 /12 = 2x 2 /3, so x = √6. We can also view the octahedron as two square pyramids with bases coincident. The square base has area x 2 = 6. The height of each pyramid is x/√2 = √3, so volume = (2/3)6√3 = 4√3. 2nd Mexican 1988 © John Scholes jscholes@kalva.demon.co.uk 30 March 2004 Last corrected/updated 30 Mar 04