Question

If a sphere is inscribed in a cube, then the ratio of the volume of the sphere to the volume of the cube is

• A. $\pi :2$
  
• B. $\pi :3$
  
• C. $\pi :4$
  
• D. $\pi :6$
  

Answer 1

The correct option is D

$\pi :6$

Let side of a cube = a

Then volume of cube $=a^3$

the diameter of inscribed sphere = a

Their radius $\left(r\right)=\frac{a}{2}$

$\therefore Volume=\frac{4}{3}\pi r^3$

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

$=\frac{4}{3}\pi \frac{a^3}{8}=\frac{\pi a^3}{6}$

$\therefore Ratiointheirvolumes=\frac{\pi a^3}{6}:a^3$

$=\frac{\pi }{6}:1=\pi :6$