Question

1. 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

• Area of signal board is 900√3 cm².

Step-by-step explanation:

Given:-

• All sides measure of triangle signal board is a.
• Perimeter of signal board is 180 cm.

To find:-

• Area of signal board.

Solution:-

Heron's formula is:-

Area of triangle = √s(s - a)(s - b)(s - c)

Where,

• s is semi-perimeter of triangle

• a, b and c are sides of triangle.

Now,

• First we will find measure of all sides of triangle using it's perimeter.

We know,

Perimeter of triangle = sum of all sides

$\implies$ a + a + a = 180

$\implies$ 3a = 180

$\implies$ a = 180/3

$\implies$ a = 60

a is measure of side.

Thus,

All sides measure of signal board is 60 cm.

Now,

• Semi-perimeter = Perimeter of triangle/2

$\implies$ Semi-perimeter = 180/2

$\implies$ Semi-perimeter = 90

Semi-perimeter of triangle is 90 cm.

So,

$\sf \longrightarrow Area = \sqrt{90(90 - 60)(90 - 60)(90 - 60)}$

$\sf \longrightarrow Area = \sqrt{90 \times 30 \times 30 \times 30}$

$\sf \longrightarrow Area = 30 \times \sqrt{3 \times (30) \times 30}$

$\sf \longrightarrow Area = 30 \times 30 \times \sqrt{3}$

$\longrightarrow$ Area = 900√3

Therefore,

Area of signal board is 900√3 cm².

Answer 2

Answer:

Given :-

• An equilateral triangle with side 'a'.
• It's perimeter is 180 cm.

To Find :-

• What is the area of the signal board.

Formula Used :-

Area of a triangle = √s(s - a)(s - b)(s - c)

Solution :-

Given :

• Perimeter = 180 cm

First, we have to find the semi-perimeter,

We know that,

↦ Semi-perimeter = Perimeter ÷ 2

↦ Semi-perimeter = 180 ÷ 2

➠ Semi-perimeter = 90 cm

Hence, all sides of an equilateral triangle are equal.

According to the question,

⇒ Perimeter = 180

⇒ a + a + a = 180

⇒ 3a = 180

⇒a = 180 ÷ 3

➦ a = 60 cm

Now, we have to use Heron's Formula,

According to the question by using the formula we get,

↣ A = √s(s - a)(s - b)(s - c)

↣ A = √90(90 - 60)(90 - 60)(90 - 60)

↣ A = √90 × 30 × 30 × 30

↣ A = √(9 × 3 × 3 × 3) × (10)⁴

↣ A = √(9²) × 3 × (10)⁴

↣ A = 9 × √3 × (10)²

↣ A = 9 × 100 × √3

➥ A = 900√3

∴ The area of the signal board is 900√3 cm².