A sphere and a cube have equal surface areas. The ratio of the volume of the sphere to that of the cube is:

Let the radius of the sphere be $r$ and the side of the cube be $a$.Given, Surface Area of Sphere = Surface Area of Cube:
$$4\pi r^2=6a^2$$
$$\frac{r^2}{a^2}=\frac{6}{4\pi }=\frac{3}{2\pi }$$
$$\frac{r}{a}=\sqrt{\frac{3}{2\pi }}$$
Now, the ratio of their volumes ($V_{\text{sphere}}:V_{\text{cube}}$):
$$\frac{V_{\text{sphere}}}{V_{\text{cube}}}=\frac{\frac{4}{3}\pi r^3}{a^3}=\frac{4}{3}\pi {\left(\frac{r}{a}\right)}^3$$
$$\frac{V_{\text{sphere}}}{V_{\text{cube}}}=\frac{4}{3}\pi {\left(\sqrt{\frac{3}{2\pi }}\right)}^3=\frac{4}{3}\pi \cdot \frac{3}{2\pi }\cdot \sqrt{\frac{3}{2\pi }}$$
$$\frac{V_{\text{sphere}}}{V_{\text{cube}}}=2\cdot \sqrt{\frac{3}{2\pi }}=\sqrt{4\cdot \frac{3}{2\pi }}=\sqrt{\frac{6}{\pi }}=\frac{\sqrt{6}}{\sqrt{\pi }}$$
Thus, the ratio is $\sqrt{6}:\sqrt{\pi }$.