Question

A traffic signal board, indicating ${}^{\prime }SCHOOLAHEA{D}^{\prime }$, is an equivalent triangle with side '$a$'. Find the area of the signal board, using Heron's formula. If its perimeter is $180cm$, what will be the area of the signal board?

Answer 1

Given:

Size of signal board $=a$

$\therefore$ Perimeter of signal board $=a+a+a=3a$

$\Rightarrow$$2s=3a,s=\frac{3}{2}a$

$\Rightarrow$$s=$ semiperimeter

Heron's formula

Area of $△=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{\left(\frac{3}{2}a\right)(\frac{3}{2}a-a)(\frac{3}{2}a-a)(\frac{3}{2}a-a)}$

$A=\sqrt{\left(\frac{3}{2}a\right){\left(\frac{a}{2}\right)}^3}=\frac{\sqrt{3}}{4}{a}^2$

$\Rightarrow$Perimeter $=180cm$ (given)

$\Rightarrow$Side$=a=\frac{180}{3}=60$ $cm$

$\Rightarrow$$A=\frac{\sqrt{3}}{4}\times 60\times 60=900\sqrt{3}$ ${cm}^2$