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

For a simple cubic unit cell:Number of atoms per unit cell $(Z)=1$.Relation between edge length ($a$) and radius ($r$): $a=2r$.Packing fraction $=\frac{\text{Volume of atoms}}{\text{Volume of unit cell}}=\frac{1\times \frac{4}{3}\pi r^3}{a^3}$.
$$\text{P.F.}=\frac{\frac{4}{3}\pi r^3}{(2r{)}^3}=\frac{\frac{4}{3}\pi r^3}{8r^3}=\frac{4\pi }{24}=\frac{\pi }{6}$$
Value is approximately $0.52$.