If the area of an equilateral triangle is

Area of equilateral triangle $=\frac{\sqrt{3}}{4} \times(\text { Side })^{2}$

$\Rightarrow \frac{\sqrt{3}}{4} \times(\text { Side })^{2}=81 \sqrt{3}$

$\Rightarrow$ (Side) $^{2}=324$

$\Rightarrow$ Side $=18 \mathrm{~cm}$

Now, we have:

Height $=\frac{\sqrt{3}}{2} \times$ Side

$=\frac{\sqrt{3}}{2} \times 18$

$=9 \sqrt{3} \mathrm{~cm}$