### Instruction: 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
$\mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(E) \frac{1}{4} }$.### Response: I need to find the volume of the octahedron and the volume of the cube, and then compare them.
The volume of the cube is easy: it's just the side length cubed.
The side length of the cube is the same as the edge length of the octahedron, so I'll call it $s$.
The volume of the octahedron is trickier: I need to divide it into two square pyramids and find the volume of each.
The base of each pyramid is a square with side length $s/\sqrt{2}$, since it is the diagonal of a face of the cube.
The height of each pyramid is the distance from the center of the cube to the center of a face, which is $s/2$.
The volume of a pyramid is one-third times the base area times the height, so the volume of each pyramid is $\frac{1}{3} \cdot \frac{s^2}{2} \cdot \frac{s}{2} = \frac{s^3}{12}$.
The volume of the octahedron is twice that, so it is $\frac{s^3}{6}$.
The ratio of the volume of the octahedron to the volume of the cube is $\frac{\frac{s^3}{6}}{s^3} = \frac{1}{6}$.
That matches answer choice (C).