2015 AMC 10B Problems/Problem 17
Problem
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?
Solution 1
The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals and , for an area . The height , of one pyramid, is , so the volume of one pyramid is . Thus, the octahedron has volume
Solution 2 (Scaling)
Because it is connected by the midpoints, the "base" of the octahedron is half the base of the rectangular prism. In addition, the volume of an octahedron is of its respective prism. Thus, the octahedron's volume is of the rectangular prism's volume, meaning that the answer is
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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