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

In simple cube, number of atoms present =$\frac{1}{8}$​$\times$$8 =1$

So, volume occupied =$1\times$$\frac{4}{3}{\mathrm{\pi r}}^3$

In simple cube atoms at corners will be touching each other.

⇒$\mathrm{a} =2\mathrm{r} ,\mathrm{r} =\frac{\mathrm{a}}{2}$

Volume of cube =${\mathrm{a}}^3$

So, fraction of total volume occupied  = $\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{\left({\displaystyle \frac{\mathrm{a}}{2}}\right)}^3}{{\mathrm{a}}^3}$

=$\frac{4}{3}\times \mathrm{\pi }\times \frac{1}{2}$

=$\frac{\mathrm{\pi }}{6}$