SOLUTION: A cube is inscribed in a sphere with diameter 9 inches. What is the volume of the cube?

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Question 589243: A cube is inscribed in a sphere with diameter 9 inches. What is the volume of the cube?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The longest distance between vertices in a cube is sqrt%283%29 times the side of the cube.

REASON:
The diagonal of one of the faces of a cube is the hypotenuse of a right triangle that has for legs the sides of that face of the cube, of length s. So the length of the diagonal of a face is
sqrt%28s%5E2%2Bs%5E2%29=sqrt%282s%5E2%29=sqrt%282%29%2As
That diagonal and the adjacent edge from another face for a right triangle, with leg lengths s and sqrt%282%29%2As. The hypotenuse is the diagonal of the cube, and its length is


If the cube is incribed in a spere of diameter 9 inches, the cube has a 9 inch diagonal and
sqrt%283%29%2As=9 --> s=9%2Fsqrt%283%29=3sqrt%283%29
The volume of a cube with a side length of s=9%2Fsqrt%283%29=3sqrt%283%29 is
V=%283sqrt%283%29%29%5E3=3%5E3%2A%28sqrt%283%29%29%5E3=27%2A3sqrt%283%29=81sqrt%283%29 cubic inches
(approximately 140.3 cubic inches).