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Volume of cube inscribed in pyramid

Source: AMC 10 2011 b Problem 22

February 24, 2011
geometry3D geometrypyramidPythagorean Theoremsimilar trianglesAMC

Problem Statement

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
<spanclass=′latex−bold′>(A)</span> 52−7<spanclass=′latex−bold′>(B)</span> 7−43<spanclass=′latex−bold′>(C)</span> 2227<spanclass=′latex−bold′>(D)</span> 29<spanclass=′latex−bold′>(E)</span> 39 <span class='latex-bold'>(A)</span>\ 5\sqrt{2}-7 \qquad <span class='latex-bold'>(B)</span>\ 7-4\sqrt{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{2\sqrt{2}}{27} \qquad <span class='latex-bold'>(D)</span>\ \frac{\sqrt{2}}{9} \qquad <span class='latex-bold'>(E)</span>\ \frac{\sqrt{3}}{9}
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