Problem:
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a cuboctahedron. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let each edge of the cube be . Then the volume of the cube is . Each of the tetrahedra removed has an isosceles right triangle of area as a base and an altitude to this base of length . Hence the volume of the cuboctahedron is
The ratio of the volume of the cuboctahedron to the volume of the cube is
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