Answer
The proof is below.
Work Step by Step
$APF=\text{{volume of atomic unit cell}}/{\text{total unit cell volume}}$
$V_{s}=\frac{4}{3}\pi r^3$
HCP has total 6 atoms, so the total volume is:
$V_{s}=6\times\frac{4}{3}\pi r^3=\frac{24}{3}\pi r^3$
For HCP, ${r}=$$\frac{a}{2}$
$V_{c}=6r^2c\sqrt{3}$
where $\frac{c}{a}=1.633$
$V_{c}=6\times3.266{\sqrt{3}}r^3$
APF =$\frac{\frac{24}{3}\pi r^3}{6\times3.266{\sqrt{3}}r^3}=0.74$
