If the area of an equilateral triangle is $81\sqrt{3}$ cm², find its perimeter.

Let the length of the side of an equilateral triangle be a cm.

$\Rightarrow \text{Area of triangle} = \dfrac{\sqrt{3}}{4} \times a^2 \\[1em] \Rightarrow 81\sqrt{3} = \dfrac{\sqrt{3}}{4} \times a^2 \\[1em] \Rightarrow a^2 = 81 \times 4 \\[1em] \Rightarrow a^2 = 324 \\[1em] \Rightarrow a = \sqrt{324} \\[1em] \Rightarrow a = 18. \\[1em]$

Perimeter = Sum of all sides of a triangle

= 3a

= 3 × 18

= 54 cm.

Hence, perimeter of the triangle = 54 cm.