The nuclear radius is of the order of ${10}^{-13}\mathrm{cm}$ while atomic radius is of the order ${10}^{-8}\mathrm{cm}.$ Assuming the nucleus and the atom to be spherical, the fraction of the atomic volume occupied by the nucleus is

The volume of a sphere $4\mathrm{\pi }{\mathrm{r}}^3/3,$ where r is the radius of the sphere.

$\begin{array}{l}\therefore \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{nucleus}=4\mathrm{\pi }{\mathrm{r}}^3/3\\ 4\mathrm{\pi }{\left({10}^{-13}\right)}^3/3{\mathrm{cm}}^3\\ \mathrm{Similarly},\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{atom}=4{\mathrm{\pi r}}^3/3=4\mathrm{\pi }{\left({10}^{-8}\right)}^3/3{\mathrm{cm}}^3\\ \mathrm{Fraction}\mathrm{of}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{atom}\mathrm{occupied}\mathrm{by}\mathrm{the}\mathrm{nucleus}\\ =\frac{4\mathrm{\pi }{\left({10}^{-13}\right)}^3/3{\mathrm{cm}}^3}{4\mathrm{\pi }{\left({10}^{-8}\right)}^3/3{\mathrm{cm}}^3}={10}^{-15}\end{array}$