A simple cubic packed crystal consists of atoms present only at the corners of the cube, touching each other at the edges.

Mathematically, $\mathrm{a}=2\mathrm{r}$

So, the volume of the unit cubic cell ${\mathrm{a}}^3={\left(2\mathrm{r}\right)}^3=8{\mathrm{r}}^3$

The volume of occupied space$=\frac{4}{3}{\mathrm{\pi r}}^3$ 

Therefore, packing fraction in a simple cubic cell$=\frac{\mathrm{Volume} \mathrm{of} \mathrm{one} \mathrm{atom}}{\mathrm{Volume} \mathrm{of} \mathrm{cubic} \mathrm{unit} \mathrm{cell}}$

$=\frac{{\displaystyle \frac{4}{3}{\mathrm{\pi r}}^3}}{8{\mathrm{r}}^3}=\frac{\mathrm{\pi }}{6}$

Hence, the required answer is A) $\frac{\pi }{6}$.