When the centers of the faces of the right rectangular prism shown below are joined to create an octahedron, what is the volume of the octahedron? $\textbf{(A) } \dfrac{75}{12} \qquad$ $\textbf{(B) } 10 \qquad$ $\textbf{(C) } 12 \qquad$ $\textbf{(D) } 10\sqrt2 \qquad$ $\textbf{(E) } 15$
Solution
For a box $a\times b\times c$, the face-center octahedron has vertices $(\pm\frac a2,0,0),(0,\pm\frac b2,0),(0,0,\pm\frac c2)$ (relative to the box's center). It consists of two pyramids with a common rhombus base with diagonals $a$ and $b$ (area $\frac{ab}2$) and heights $\frac c2$, so its volume is $2\cdot\frac13\cdot\frac{ab}2\cdot\frac c2=\frac{abc}6$.
With $a=4,b=5,c=3$: volume $=\frac{4\cdot5\cdot3}6=10$. The answer is (\textbf{B}).
$\textbf{(A) } \dfrac{75}{12} \qquad$ $\textbf{(B) } 10 \qquad$ $\textbf{(C) } 12 \qquad$ $\textbf{(D) } 10\sqrt2 \qquad$ $\textbf{(E) } 15$ 
