The fraction of the total volume occupied by the atoms in a simple cube is

In a simple cubic structure, the atoms are present at all the corners. The effective number of atoms inside a unit cell is one.

Thus, the volume occupied by the atoms $=\frac{4}{3}{\mathrm{\pi r}}^3 \left(\mathrm{r}=\text{radius of the atom}\right)$

The volume of the cube $={\mathrm{a}}^3 \left(\text{where}, \mathrm{a}=\text{side length}\right)$

The relation between side length and the radius of the atoms$\Rightarrow 2\mathrm{r}=\mathrm{a}$

Thus, the fraction of total volume occupied by the atoms$=\frac{4{\mathrm{\pi r}}^3}{3{\mathrm{a}}^3}=\frac{4\mathrm{\pi }}{3}{\left(\frac{1}{2}\right)}^3=\frac{\mathrm{\pi }}{6}$