Packing fraction in F.C.C. lattice is :

The number of atoms in a face-centered cubic unit cell is four. A line that runs diagonally from a cube's top corner to its bottom corner on the same side of the cube is equivalent to 4r. A and r can be connected by using geometry and the side length as follows:

$a=2r\sqrt{2}$

$\text{ Packing fraction }=\frac{\text{ Number of atom }\times \text{ volume of atom }}{\text{ volume of unit cell }}$

$\Rightarrow \frac{4\times \frac{4}{3}\pi r^3}{(2\sqrt{2}r{)}^3}$

$\therefore \text{ Packing fraction }=\frac{\pi \sqrt{2}}{6}$