sphere and inscribed cube We can express ratio between the volume of a cube and volume of inscribed sphere  in this cube as $6/\pi$ . What is the ratio between these objects  when the cube is inscribed in a sphere?
Consider the cube within the sphere. It has sides $s$ and long diagonal $2a$, where $a$ is the radius of the sphere. Applying the Pythagorean theorem twice, we can show that $s=2a/\sqrt{3}$. It then follows that

$$\frac{V_{sphere}}{V_{cube}}=\frac{\frac{4}{3}\pi a^3}{\frac{8a^3}{3\sqrt{3}}}=\frac{\sqrt{3}\pi}{2}\approx 2.7207$$
I've calculated, sphere:incribed cube = 2.7206990463:1
Since its about ratio, we can make the diameter of the sphere anything, so I took 2. This makes the space diagonal of the cube=2. The sides of the cube are thus 2/root3. The volume of cube= 8root3/9. The volume of the sphere is 4(pi)r³/3 ad in this case, 4(pi)/3. If you take these two numbers (4(pi)/3 and 8root3/9), the ratio of sphere to inscribed cube is 2.7206990463:1