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

In a simple cubic arrangement, the number of atoms is 1. 

Also, a = 2r

Therefore, packing fraction 

             $\mathrm{P}.\mathrm{F} =\frac{\mathrm{volume} \mathrm{occupied} \mathrm{by} \mathrm{particles}}{\mathrm{volume} \mathrm{of} \mathrm{unit} \mathrm{cell}}$

$$\begin{gathered} \text{ =}\frac{\stackrel{4}{}/3{\mathrm{\pi r}}^3\times \mathrm{Z}}{{\mathrm{a}}^3} \\ \mathrm{But} \mathrm{Z} = 8\times \frac{1}{8}=1 \\ =\frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3\times 1}{{\left(2\mathrm{r}\right)}^3} \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$