Question

A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side ${}^{\prime }{a}^{\prime }$. 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?

• A. $900c{m}^2$
• B. $900\sqrt{3}c{m}^2$
  
• C. $900\sqrt{2}c{m}^2$
  
• D. $2700c{m}^2$

Answer 1

The correct option is B $900\sqrt{3}c{m}^2$

Area =$\sqrt{s(s-a)(s-b)(s-c)}$ where, $s=\frac{a+b+c}{2}$

In an equilateral triangle, $s=\frac{3a}{2}$

$\Rightarrow$ Area $=\sqrt{\frac{3a}{2}{(\frac{3a}{2}-a)}^3}$

$=\sqrt{\frac{3a^4}{2^4}}$

$\Rightarrow$ Area $=\frac{\sqrt{3}}{4}{a}^2$

Given perimeter $=180$ cm

$\Rightarrow 3a=180$

$\Rightarrow a=60$ cm

$\Rightarrow$ Area $=\frac{\sqrt{3}}{4}\times {60}^2=900\sqrt{3}c{m}^2$