Problem:
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of the octahedron?
Answer Choices:
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Solution:
Consider the octahedron to be two pyramids whose base is a rhombus in the middle horizontal plane, as shown below. One pyramid points up, the other down. The area of the base is the area of right triangles with legs and , or . The altitude of each pyramid is half that of the prism or . The volume of the octahedron is .
The problems on this page are the property of the MAA's American Mathematics Competitions