If the area of an equilateral triangle is

Area of the equilateral triangle $=81 \sqrt{3} \mathrm{~cm}^{2}$

Area of an equilateral triangle $=\left(\frac{\sqrt{3}}{4} \times a^{2}\right)$, where $\mathrm{a}$ is the length of the side.

$\Rightarrow 81 \sqrt{3}=\frac{\sqrt{3}}{4} \times a^{2}$

$\Rightarrow 324=a^{2}$

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

Height of triangle $=\frac{\sqrt{3}}{2} \times a$

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

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