Calculate the packing fraction for equilateral riangular rods stacked as shown.

Let A be area of larger tnangle.

Let $\alpha$ be area of smaller tnangle

Base of larger triangle = B

Base of smaller triangle, $b=\frac{1}{2}B$

Height of larger tnangle = H

Height of smaller triangle, $\mathrm{h}=\frac{1}{2}\mathrm{H}$

$\Rightarrow \mathrm{a}=\frac{1}{2}\times \mathrm{b}\times \mathrm{h}=\frac{1}{4}\mathrm{A}$

Let total no. of larger triangles be "N". From the figure it is clear that on an average each larger triangle is associated with two smaller triangles.

Area of white space$=\frac{2\mathrm{NA}}{4}$

Total space $=NA+\frac{NA}{2}=\frac{3NA}{2}$

$\therefore \eta =\frac{\text{ Area of shaded triangle }}{\text{ Total area }}=\frac{NA}{\frac{3NA}{2}}=\frac{2}{3}$