A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?

We know that,

Each side of the equilateral triangle is equal.

Given,

Length of each side of an equilateral triangle = a cm.

Perimeter of traffic signal board (equilateral triangle) = sum of all the sides = a + a + a = 3a cm.

By formula,

Semi perimeter (s) = $\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{3a}{2}$ cm.

By Heron's formula,

Area of triangle (A) = $\sqrt{s(s - a)(s - b)(s - c)}$ sq.units, where a, b and c are sides of triangle.

Substituting values we get :

$A = \sqrt{\dfrac{3a}{2} \times \Big(\dfrac{3a}{2} - a\Big) \times \Big(\dfrac{3a}{2} - a\Big) \times \Big(\dfrac{3a}{2} - a\Big)} \\[1em] = \sqrt{\dfrac{3a}{2} \times \Big(\dfrac{3a - 2a}{2}\Big) \times \Big(\dfrac{3a - 2a}{2}\Big) \times \Big(\dfrac{3a - 2a}{2}\Big)} \\[1em] = \sqrt{\dfrac{3a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2}} \\[1em] = \sqrt{\dfrac{3a^4}{16}} \\[1em] = \dfrac{\sqrt{3}}{4}a^2.$

Given,

Perimeter = 180 cm

∴ 3a = 180

⇒ a = $\dfrac{180}{3}$ = 60 cm.

Substituting value of a, we get :

$\text{Area of triangle (A)} = \dfrac{\sqrt{3}}{4}a^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 60^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 3600 \\[1em] = 900\sqrt{3}\text{ cm}^2.$

Hence, the area of the signal board is $900\sqrt{3}$ cm².