Question

What is the ratio of the volume of a cube to that of the sphere which will fit inside it?

Answer 1

The sphere which exactly fits into the cube will have the diameter as the same as the side of the cube.

If $l$ is the side of the cube , then

Volume of the cube $V_1=l^3$

Volume of the sphere $V_2=\frac{4}{3}\pi {\left(\frac{l}{2}\right)}^3$

$\therefore V_1:V_2$ is $l^3:\frac{4}{3}\pi {\left(\frac{l}{2}\right)}^3$

$\Rightarrow l^3=\frac{\pi l^3}{6}$

$\therefore$ the ratio is $6:\pi$