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

Side of the traffic signal board = a
Perimeter of traffic signal board = $3\times a$
$2s=3a\Rightarrow s=\frac{3}{2}a$
By Heron’s formula,
Area of the triangle
= $\sqrt{s(s-a)(s-b)(s-c)}$
= $\sqrt{\frac{3}{2}a(\frac{3}{2}a-a)(\frac{3}{2}a-a)(\frac{3}{2}a-a)}=\sqrt{\left(\frac{3}{2}a\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)}=\frac{\sqrt{3}}{4}{a}^2\dots \dots \left(1\right)Perimeterofthetrafficsignalboard=180cmSideofthetrafficsignalboard\left(a\right)=\left(\frac{180}{3}\right)cm=60cmUsingequation\left(1\right),areaofthetrafficsignalboard=\frac{\sqrt{3}}{4}(60cm{)}^2=\left(\frac{3600}{4}\sqrt{3}\right)c{m}^2=900\sqrt{3}c{m}^2$