2009 AMC 12A Problem 22

Question

A regular octahedron has side length

1.

A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area

\dfrac{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?

10

11

12

13

14

Answer: E

Concepts:3D geometry regular polygon area

Difficulty rating: 2270

Small hint:

The cutting plane passes through the midpoints of the six edges that do not belong to the two parallel faces

Big hint:

Those midpoints form a regular hexagon with side

\dfrac{1}{2};

find its area

Solution:

Let the two parallel faces be triangles. The plane passes through the midpoints of the six edges not on those faces, forming an equilateral hexagon of side

\dfrac{1}{2},

which by symmetry is also equiangular and hence regular.

A regular hexagon is six equilateral triangles, so its area is

6\cdot\frac{\sqrt{3}}{4}\left(\frac{1}{2}\right)^2 = \frac{3\sqrt{3}}{8}.

Thus

a = 3,

b = 3,

c = 8,

and

a + b + c = 14.

Thus, the correct answer is E.