AMC12 2009 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 $a + b + c$?

Solution: The plane cuts the octahedron into two congruent solids, which allows us to consider only the cut through the top half (a square pyramid). Because the cut is parallel to one side of the pyramid and must create two congruent solids, we can see that it must take the shape of a trapezoid with right, left, and top sides being 0.5 and a bottom side of 1. This is equivalent to a unit equilateral triangle with the tip cut off at the midpoint. Accordingly, the area of the polygon is $2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \cdot \frac{3}{4}$ or $\frac{3\sqrt{3}}{8}$, and $a + b + c = 14$. $\fbox{14}$.

Answer: 14