2015 AMC 10B Problem 17

Question

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?

\dfrac{75}{12}

10

12

10\sqrt{2}

15

Answer: B

Concepts:polyhedron pyramid volume

Difficulty rating: 1420

Small hint:

View the octahedron as two pyramids with a central rhombus base

Big hint:

The base rhombus has diagonals

4

and

5

Solution:

The octahedron can be viewed as two congruent pyramids whose shared base is the rhombus through the centers of the four side faces. This rhombus has diagonals

4

and

5

, so its area is

\frac12\cdot4\cdot5=10

.

Each pyramid has height

\frac32

, half the prism’s height. Thus the total volume is

2\cdot\frac13\cdot10\cdot\frac32=10.

Thus, the correct answer is B.