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Area of volume formed by intersection of tetrehedra

Source: 2011 AMC A Problem 24

June 25, 2011
geometry3D geometrytetrahedronratiooctahedronrhombusUSAMTS

Problem Statement

Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
<spanclass=′latex−bold′>(A)</span> 112<spanclass=′latex−bold′>(B)</span> 212<spanclass=′latex−bold′>(C)</span> 312<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 26 <span class='latex-bold'>(A)</span>\ \frac{1}{12}\qquad<span class='latex-bold'>(B)</span>\ \frac{\sqrt{2}}{12}\qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt{3}}{12}\qquad<span class='latex-bold'>(D)</span>\ \frac{1}{6}\qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt{2}}{6}
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