Question

A traffic signal board, indicating 'SCHOOLAHEAD', is an equilateral 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?

• A. $900\sqrt{3}c{m}^2$
• B. $300\sqrt{3}c{m}^2$
• C. $600\sqrt{3}c{m}^2$
• D. $900\sqrt{2}c{m}^2$

Answer 1

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

Area of triangle $=\sqrt{S(S-a)(S-b)(S-c)}$

Perimeter $=180cm$

Semi-perimeter, $S=\frac{a+b+c}{2}=\frac{180}{2}=90cm$

Given triangle is an equilateral triangle, therefore $a=b=c$

Perimeter $=3a$

$180=3a\Rightarrow a=60$

$a=b=c=60cm$

Area of triangle

$=\sqrt{S(S-a)(S-b)(S-c)}$
$=\sqrt{90(90-60)(90-60)(90-60)}$
$=\sqrt{90\times 30\times 30\times 30}$
$=\sqrt{9\times 3\times 3\times 3\times {\left(10\right)}^4}$
$=3\times 3\times {10}^2\sqrt{3}$
$=900\sqrt{3}c{m}^2$