Hi Justin.

Let's say the radius of the sphere is r. Then you can find an expression for the volume of the sphere, right?

If the cube is centered in the sphere (we call it "inscribed"), then the center of the cube to any one of its corners is r.

Use Pythagorus two times to relate the side length of the cube to the inner half-diagonal (the radius r). Then cube the side length to find the volume of the cube.

The ratio of the two volumes gives you the percentage you seek.

Cheers,
Stephen La Rocque.