A sphere and a cube have equal surface areas.

Surface area of sphere = sphere area of cube

i.e., $4 \pi r^{2}=6 a^{2}$

$\frac{r^{2}}{a^{2}}=\frac{6}{4 \pi}$

$\frac{r}{a}=\left(\frac{6}{4 \pi}\right)^{\frac{1}{2}} \ldots \ldots$(1)

Now, $\frac{\text { volume of sphere }}{\text { volume of cube }}=\frac{\frac{4}{3} \pi r^{3}}{a^{3}}$

$\frac{v_{1}}{v_{2}}=\frac{4 \pi r^{3}}{3 a^{3}}$

$=\frac{4}{3} \pi\left(\frac{r}{a}\right)\left(\frac{r}{a}\right)^{2}$

$=\frac{4}{3} \pi \sqrt{\frac{6}{\pi}} \frac{1}{2} \frac{6}{4 \pi}$

$\frac{v_{1}}{v_{2}}=\sqrt{\frac{6}{\pi}}$

$v_{1}: v_{2}=\sqrt{\frac{6}{\pi}}$