The largest cube inside a sphere

Question

A sphere has radius 1. What is the side length of the largest cube that fits inside the sphere? What fraction of the sphere's volume does that cube occupy? Try to answer the first part with a single application of Pythagoras rather than any calculus, and give the fraction as a rough percentage.

Accepted Answer

Side length 2 divided by the square root of 3, about 1.155. The cube fills about 37 per cent of the sphere. The diagonal. For a cube of side s, the diagonal across one face has length s times root 2 (Pythagoras in the face). The long diagonal through the cube's centre has length the square root of (s squared + s squared + s squared) = s times root 3. The largest inscribed cube has all eight corners on the sphere, so its long diagonal is a diameter, 2. Hence s root 3 = 2 and s = 2 / root 3, about 1.155. The volume fraction. The cube's volume is s cubed = 8 / (3 root 3), about 1.540. The sphere…

The largest cube inside a sphere

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