Consider the top square ABCD of the cube. The centre of this square is at point P and P touches the inscribed sphere. The perpendicular from P to AB is the same as the radius r of the inscribed sphere. Perpendicular to the plane of ABCD and a distance r below it is O, the centre of the sphere and cube.
So OP=r and PB=r√2, being a semi-diagonal of ABCD which has a side length of 2r. In the right triangle OPB, OP=r, PB=r√2 and OB, the hypotenuse=√(OP²+PB²)=√(r²+2r²)=√(3r²). But OB is the radius of the bigger circumscribed sphere. Therefore the volume of this sphere is (4/3)π(√(3r²))³.
Putting in r=3m, we get the volume (4/3)π(√27)³ or (4/3)π(27)^(3/2)=587.671 cubic metres approx.