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

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.

2015 AMC 10B Problem 17

Answer: B

Difficulty rating: 1420

Solution:

Problem 17 in Other Years