Sphere inscribed in a cube

Statement

A sphere is inscribed in a cube with side $a$. Calculate the ratio $V_{\text{cubo}}/V_{\text{sfera}}$.

Solution

The inscribed sphere has diameter equal to the side of the cube, hence radius $R = a/2$. $V_{\text{cubo}} = a^3, \qquad V_{\text{sfera}} = \tfrac{4}{3}\pi R^3 = \tfrac{4}{3}\pi\left(\tfrac{a}{2}\right)^3 = \frac{\pi a^3}{6}.$ The ratio is therefore $\frac{V_{\text{cubo}}}{V_{\text{sfera}}} = \frac{a^3}{\pi a^3/6} = \boxed{\dfrac{6}{\pi}} \approx 1{,}91.$

Links

Topics: Synthetic geometry in space
Concepts: Parallelepiped · Sphere · Volume
Skills: Calculating · Using formulae
Exercise types: Geometric problem