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 $\sqrt{6}:\sqrt{\mathrm{\pi }}$.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{the}\mathrm{sphere}\mathrm{be}r\mathrm{and}\mathrm{the}\mathrm{edge}\mathrm{of}\mathrm{the}\mathrm{cube}\mathrm{be}a. \\ \mathrm{As}, \\ \mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{sphere}=\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cube} \\ \Rightarrow 4\mathrm{\pi }{r}^2=6a^2 \\ \Rightarrow \frac{r^2}{a^2}=\frac{6}{4\mathrm{\pi }} \\ \Rightarrow {\left(\frac{r}{a}\right)}^2=\frac{3}{2\mathrm{\pi }} \\ \Rightarrow \frac{r}{a}=\sqrt{\frac{3}{2\mathrm{\pi }}}.....\left(\mathrm{i}\right) \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{sphere}\mathrm{to}\mathrm{that}\mathrm{of}\mathrm{the}\mathrm{cube}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cube}} \\ =\frac{\left({\displaystyle \frac{4}{3}\mathrm{\pi }{r}^3}\right)}{a^3} \\ =\frac{4}{3}\mathrm{\pi }{\left(\frac{r}{a}\right)}^3 \\ =\frac{4}{3}\mathrm{\pi }{\left(\sqrt{\frac{3}{2\mathrm{\pi }}}\right)}^3\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ =\frac{4}{3}\mathrm{\pi }\left(\frac{3\sqrt{3}}{2\mathrm{\pi }\sqrt{2\mathrm{\pi }}}\right) \\ =\frac{2\sqrt{3}}{\sqrt{2\mathrm{\pi }}} \\ =\frac{2\sqrt{3}\times \sqrt{2}}{2\sqrt{\mathrm{\pi }}} \\ =\frac{\sqrt{6}}{\sqrt{\mathrm{\pi }}} \\ =\sqrt{6}:\sqrt{\mathrm{\pi }} \end{gathered}$$

So, the ratio of the ratio of the volume of the sphere to that of the cube is $\sqrt{6}:\sqrt{\mathrm{\pi }}$.