Let the side of the inscribed cube be 2a. The centre of the cube and the centre of the sphere coincide. The radius of the sphere is also the length of a line from the centre of the city be to a vertex. To find the radius we imagine looking down on to its top face. Draw a line from the centre of this face to a corner; this is a semi diagonal and has length sqrt(2a^2)=asqrt(2). This is the base of a right-angled triangle with base asqrt(2), height a (half of one side of the cube) and hypotenuse = radius of the sphere, r. So r=sqrt(2a^2+a^2)=asqrt(3) and the volume of the sphere is:
4(pi)(asqrt(3))^3/3=(4/3)(pi)a^3*3^(3/2). The volume of the cube is 8a^3, so the ratio of the volumes is (3/4(pi)*3^(3/2))=0.36755 approx. or 6:(pi)*3^(3/2) or 1:2.72 approx.