The largest possible cube is cut from a solid sphere of radius 6√3 cm. Find the edge of the cube in cm.

The cube is as large as possible when its four space diagonals become diameters of the sphere, so its diagonal = $2\times 6$√3 = 12√3 cm. Since the diagonal of a cube of edge a is a√3, a√3 = 12√3 gives $a=12$ cm. (Taking the diameter itself as the edge gives 12√3 ≈ 20.8 cm, a cube far too big to fit inside.)