+15 A cube has side length . Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)
+1667
A cube has side length . Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)
"A cube has side length ." Hard to solve a problem without that piece of information.
Let's assign our own length to the side of the cube. I think 1 is a pretty good number.
You'll note that every edge of the tetrahedron is the diagonal of a side of the cube.
If 1 is the length of the cube side, then sqrt(2) is the length of the tetrahedron edge.
The volume of a regular tetrahedron in terms of the length of its edge is given as:
V = a3 / 6 • sqrt(2) where a is the length of the edge of the tetrahedron
V = sqrt(2)3 / 6 • sqrt(2) divide that sqrt(2) in the denominator into the numerator
V = sqrt(2)2 / 6
V = 2 / 6 = 1 / 3
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