Fraction of Space Filled by Hexagonal Close Packing

Suppose spheres of radius  
  are arranging in a hexagonal close packing arrangement. We can take a unit cell as a a sphere with a sphere at each of the eight vertices (shared between eight other unit cells) and a sphere at the centre. The total volume occupied by the spheres is
\[8 (\frac{4/3 \pi r^3}{8}) + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3\]

The diagonal of the unit cell is  
  so if so taking the unit cell to have side  
  and using Pythagoras Theorem in three dimensions,  
\[a^2+a^2+a^2=(4r)^2 \rightarrow a = \sqrt{4r}{\sqrt{3}}\]

The fraction of the cell that is occupied by the spheres is  
\[\frac{8/3 \pi r^3}{(4r/ \sqrt{3}})^3 =\frac{\pi \sqrt{3}}{8}\]

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