The surface area of a cube is equal to the surface area of a sphere. The ratio of their volumes will be

The correct option is D: $\sqrt{\pi }:\sqrt{6}$

Total surface area of the cube if its side ${}^{\prime }{a}^{\prime }$ units will be $6a^2$

Surface area of the sphere if its radius is ${}^{\prime }{r}^{\prime }$ units will be $4\pi r^2$

Given: Surface area of cube $=$ Surface area of sphere

$\Rightarrow 6a^2=4\pi r^2$

$\Rightarrow [\frac{a}{r}{]}^2=\frac{2}{3}\pi$

$\Rightarrow \frac{a}{r}=\sqrt{\frac{2}{3}\pi }$

Volume of the cube $=a^3$

and, Volume of sphere $=\frac{4}{3}\pi r^3$

$\therefore$ Ratio of their volumes $=\frac{a^3}{\frac{4}{3}\pi r^3}$

$=\frac{3}{4\pi }\times \frac{a^3}{r^3}$

$=\frac{3}{4\pi }[\sqrt{\frac{2}{3}}\pi {]}^3$

$=\frac{3}{4\pi }\times \frac{2}{3}\times \pi \sqrt{\frac{2}{3}\pi }$

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

Hence, the ratio of their volumes is $\sqrt{\pi }:\sqrt{6}$