 Hard-sphere cubic packing fractions

Nearest touching spheres on an FCC conventional cell meet along face diagonals, giving $4r=\sqrt{2}a$ and four spheres per cell. In BCC they meet along body diagonals, giving $4r=\sqrt{3}a$ and two spheres per cell. Dividing the respective sphere volumes by $a^3$ proves the displayed packing fractions. A coordination-number comparison alone does not calculate the occupied volume.