1989 AHSME Problems/Problem 26
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
Call the length of a side of the cube x. Thus, the volume of the cube is . We can then find that a side of this regular octahedron is the square root of + which is equivalent to . Using our general formula for the volume of a regular octahedron of side length a, which is , we get that the volume of this octahedron is...
Comparing the ratio of the volume of the octahedron to the cube is…
$\frac{\frac{x^3}{6}}{x^3} \rightarrow \framebox[1.1\width]{(C) \frac{1}{6} }$ (Error compiling LaTeX. Unknown error_msg) The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.