Question

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?

Answer 1

Let each side of the equilateral triangle be a.
Semi-perimeter of the triangle,
$S=\frac{a+a+a}{2}=\frac{3a}{2}$
Area of the triangle = $\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{s(s-a)(s-a)(s-a)}=\sqrt{s(s-a{)}^3}\sqrt{\frac{3a}{2}(\frac{3a}{2}-a{)}^3}\sqrt{\frac{3a}{2}(\frac{a}{2}{)}^3}=\sqrt{\frac{3a^4}{2^4}}=\frac{\sqrt{3}}{4}{a}^2$
Now, its perimeter is 180 cm.
$\Rightarrow$ a+a+a= 180 cm
$\Rightarrow$ 3a = 180 cm
$\Rightarrow$ a = $\frac{180}{3}$ cm = 60 cm
Thus, the area of triangle is =$\frac{\sqrt{3}}{4}{a}^2=\frac{\sqrt{3}}{4}(60{)}^2=900\sqrt{3}c{m}^2$