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

We know that,

Area of equilateral triangle = $\dfrac{\sqrt{3}}{4}$(side)²

Substituting the values,

$\Rightarrow 81\sqrt{3} = \dfrac{\sqrt{3}}{4}(side)^2 \\[1em] \Rightarrow (side)^2 = \dfrac{81\sqrt{3} \times 4}{\sqrt{3}} \\[1em] \Rightarrow (side)^2 = 81 \times 4 \\[1em] \Rightarrow (side)^2 = 324 \\[1em] \Rightarrow \text{side} = \sqrt{324} = 18 \text{ cm}.$

So the perimeter of equilateral triangle = 3 × side

= 3 × 18 = 54 cm.

Hence, perimeter of equilateral triangle = 54 cm.