Calculate packing fraction of hexagonal packing of circles in two dimension.$\text{ [Given : }\left.\frac{\pi }{3\sqrt{3}}=0.600\right]$

Express your answer after multiplying your answer with 1000.

$\text{ Packing fraction }=\frac{\text{ Area occuhied hyatoms }}{\text{ Area of hexagon }}$

$=\frac{3\pi r^2}{6\sqrt{3}{r}^2}$

$=\frac{1}{2}\times \frac{\pi }{\sqrt{3}}$        ......(i)

$$\begin{gathered} \mathrm{Given}: \\ \frac{\pi }{3\sqrt{3}}=0.600 \\ \frac{\pi }{\sqrt{3}}=1.800 \end{gathered}$$

Put the value in equation (i),

$$\begin{gathered} =\frac{1}{2}\times 1.800 \\ =0.900 \end{gathered}$$

When multiplied with 1000.

$$\begin{gathered} =0.900\times 1000 \\ =900 \end{gathered}$$