Sphere inscribed in a cube: volume ratio

Text

A sphere is inscribed in a cube of edge $10$ cm. Find the ratio between the volume of the cube and the volume of the sphere.

Solution

The inscribed sphere has radius $r=\dfrac{\text{edge}}{2}=5$ cm. $V_\text{cube}=10^3=1000$; $\ V_\text{sphere}=\dfrac43\pi\cdot125=\dfrac{500\pi}{3}$. $\frac{V_\text{cube}}{V_\text{sphere}}=\frac{1000}{\tfrac{500\pi}{3}}=\frac{6}{\pi}\approx1.91.$ $\boxed{\dfrac{V_\text{cube}}{V_\text{sphere}}=\dfrac6\pi\approx1.91}$