Question:
If the area of an equilateral triangle is $81 \sqrt{3} \mathrm{~cm}^{2}$, find its height.
Solution:
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 $=\frac{\sqrt{3}}{2} \times a$
$=\frac{\sqrt{3}}{2} \times 18$
$=9 \sqrt{3} \mathrm{~cm}$