Question

A sphere and a cube have the same surface area. Show that the ratio of the volume of the sphere to that of the cube is $\sqrt{6}:\sqrt{\pi }$

Answer 1

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

Given, that sphere and cube have same surface area,

So,

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

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

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

$\Rightarrow \frac{r}{a}=\sqrt{\frac{3}{2\pi }}.......\left(1\right)$

Now,

Ratio of their volumes

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

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

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

$=\frac{4\pi }{3}\times {\left(\sqrt{\frac{3}{2\pi }}\right)}^3$

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

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

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

Hence proved.