Question

A sphere and a cube have the same surface area. Find out the ratio of the volume of sphere to that to the cube.

• A. $6:\pi$
• B. $\sqrt{6}:\sqrt{\pi }$
• C. $\sqrt{6}:\pi$
• D. $6:\sqrt{\pi }$

Answer 1

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

Let radius of sphere $=r$ and Side of cube $=a$, then

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

Ratio of Volumes $=\frac{\frac{4}{3}\pi r^3}{a^3}=\frac{4}{3}\pi {\left(\frac{r}{a}\right)}^3=\frac{4}{3}\pi \times \frac{6}{4\pi }\times \frac{1}{2}\sqrt{\frac{6}{\pi }}=\sqrt{6}:\sqrt{\pi }$