2009 AMC 12A 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\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.