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

Given :–

• A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a.
• If its perimeter is 180 cm.

To Find :–

1. Find the area of the signal board, [Using heron's formula]
2. what will be the area of the signal board ?

Solution :–

We know that,

→ s = $\sf \frac{a + b + c}{2}$

According to the question,

The board is an equilateral triangle.

It means,

• a = b = c.

So,

→ s = $\sf \frac{a + a + a}{2}$

• s = $\sf \frac{3a}{2}$

Area of the signal board,

→ $\sqrt{s(s - a)(s - b)(s - c)}$

Hence,

• a = b = c

So,

→ $\sqrt{ \frac{3a}{2}( \frac{3a}{2} - a)( \frac{3a}{2} - a)( \frac{3a}{2} - a) }$

→ $\sqrt{ \frac{3a}{2}( \frac{3a}{2} - \frac{a}{1} )( \frac{3a}{2} - \frac{a}{1} )( \frac{3a}{2} - \frac{a}{1} ) } →$

Take the L.C.M. of the denominator (2 and 1) = 2.

→ $\sqrt{ \frac{3a}{2}( \frac{a}{2})( \frac{a}{2} )( \frac{a}{2} ) }$

→ $\frac{a}{2} \times \frac{a}{2} \sqrt{3}$

→ $\frac{ {a}^{2} }{4} \sqrt{3}$

→ $\frac{ \sqrt{3} }{4} {a}^{2}$

Now, it's perimeter is 180cm.

Perimeter of an equilateral = 180cm

• Side = a

→ 3a = 180cm

→ a = $\frac{\cancel{180}}{\cancel3}$

→ a = 60cm

• a = 60cm.

Now, area of the signal board = $\frac{\sqrt{3}}{4} {a}^{2}$

→ $\frac{\sqrt{3}}{4} {(60)}^{2}$

→ $\frac{\sqrt{3}}{\cancel4} \times 60 \times \cancel{60}$

→ $\frac{\sqrt{3}}{\cancel2} \times 60 \times \cancel{30}$

→ √3 × 60 × 15

→ √3 × 900

→ 900√3 cm²

Hence,

1. Area of the signal board = $\frac{\sqrt{3}}{4} {a}^{2}$.
2. Area of singal board = 900√3 cm².