AMC 12 2009a Problem 22

A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?

\textbf{(A)}\ 10 \textbf{(B)}\ 11 \textbf{(C)}\ 12 \textbf{(D)}\ 13 \textbf{(E)}\ 14