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?

Here, the side of triangle are a, a and units.
So, the semi-perimeter of triangle is given by s = a+a+a/2 = 3a/2
Therefore, using Heron’s formula, the area of triangle = √s(s – a) (s – b) (s – c)
= √((3a/2(3a/2) – a)(3a/2) – a)) = √((3a/2(3a – 2a/2)(3a – 2a/2)(3a – 2a/2))
= √3a/2(a/2)(a/2)(a/2) = (a²/4)√3
Perimeter of equilaterral triangle = 3a
According to question, 3a = 180 cm ⇒ a = 180/3 = 60 cm
Therefore, the area of triangle = (a²/4)√3 = ((60)²/4))√3 = (3600/4)√3 = 900 √3 cm².