
In order to make proper calculations, let's
rotate the small (green) cube in the sphere first in such a way as it
is shown in the illustration. Now it can be seen clearly the small
cube's diagonal is equal the side of the big (orange) cube.
The following calculations provide us with the final volume of the
small (green) cube.
Volume of the Orange Cube = 1;
Edge of the Orange Cube = 1;
Main Diagonal of the Green Cube = 1;
(Edge of the Green Cube)^{2} + (Edge of the Green Cube)^{2}
+ (Edge of the Green Cube)^{2} = 1^{2};
3*(Edge of the Green Cube)^{2} = 1;
(Edge of the Green Cube)^{2} = 1/3;
Edge of the Green Cube = 1/sqrt(3) = sqrt(3)/3;
Volume of the Green Cube = [sqrt(3)/3]^{3} = sqrt(3)/9 =
0.193. 
