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?

Given:

The side of an equilateral triangle $=a$.
Perimeter of the triangle $=180\ cm$.

To do:

We have to find the area of the signal board.

Solution:

Perimeter $=a+a+a=180\ cm$

$3a=180\ cm$

$a=\frac{180}{3}\ cm$

$a=60\ cm$

Semi-perimeter $=\frac{180}{2}=90$

Area of a triangle with sides $a, b, c$ and semi-perimeter $s$ is given by $\sqrt{s(s-a)(s-b)(s-c)}$    (Heron's formula)

Therefore,

Area of the signal board $=\sqrt{90(90-60)(90-60)(90-60)}$

$=\sqrt{90(30)(30)(30)}$

$=\sqrt{9(3)(3)(3)\times10^4}$

$=3\times3\times10^2\sqrt3$

$=900\sqrt{3}\ cm^2$

The area of the signal board is $900\sqrt3\ cm^2$.