= Hard-sphere cubic packing fractions
{title2=$\eta_{\rm FCC}=\pi/(3\sqrt2),\quad\eta_{\rm BCC}=\sqrt3\pi/8$}
Nearest touching spheres on an FCC conventional cell meet along face diagonals, giving $4r=\sqrt2a$ and four spheres per cell. In BCC they meet along body diagonals, giving $4r=\sqrt3a$ 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.
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