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Tetrahedra Trial (Posted on 2014-08-27) Difficulty: 3 of 5
Two distinct regular tetrahedra entirely contained within an unit cube - have all their vertices among the vertices of the same unit cube.

Determine the volume of the region formed by the intersection of the tetrahedra.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 1 of 1
The intersetion of the tetrahedra is a regular octahedron with each of its six vertices at the center of one of the faces of the cube.

The octahedraon can be considered as consisting of two square-based pyramids with equilateral slant faces. As the opposite pairs of vertices are on opposite faces of the cube, their separations are 1 unit each, and the base-to-base pyramids each have height 1/2.

The edges of the pyramid, both slant and along the square base are going from the center of a square face of the original cube to the center of an adjacent face, and have the length sqrt(2)/2. Thus the square base of each pyramid had area 2/4 = 1/2.

The volume of each pyramid is then (1/2)*(1/2)/3 = 1/12.

Since the octahedron in question consists of two of these pyramids, it has volume 1/6.

  Posted by Charlie on 2014-08-27 19:39:03
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