Question

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

Answer 1

Let $r$ be the radius of the sphere and $a$ be the side of the cube.

According to the question,

Surface area of cube $=$ Surface area of sphere

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

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

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

Therefore,

Volume of sphere $:$ Volume of cube $=\frac{\frac{4}{3}\pi (\sqrt{3}{)}^2}{(\sqrt{2\pi }{)}^3}$

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

$=\frac{\sqrt{6}}{\sqrt{\pi }}$