The largest 'rectangular box' will be a cube.
For any rectangular box, with dimensions: L,W,H
The length of the diagonal is
The diagonal of the cube inside the sphere is equal to the diameter of the sphere, or twice the radius.
Since the sides of the cube are equal, lets call it 's'
and the diameter of the sphere = 2r
The Volume of the cube is s^3
substituting the value above for s
Volume =
.