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

Hint: Firstly use the formula “Perimeter of equilateral triangle = 3 (Side)”, you will get the value of ‘a’. Then find the value of ‘s’ by using the formula $s=\dfrac{Perimeter}{2}$. At the end use the Heron’s Formula “Area of triangle $=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$” to get the final answer.

Complete step-by-step answer:

To solve the given question we will write the given data first, therefore,
Side of the triangle = a (1)
Perimeter of triangle = 180 cm (2)
As the triangle is equilateral therefore we should know the formula of perimeter of equilateral triangle given below,
Formula:
Perimeter of equilateral triangle = 3 (Side)
By using the above formula we can write the formula for perimeter of equilateral triangle with side ‘a’ as given below,
Perimeter of triangle = 3a
If we compare above equation with equation (2) we will get,
Therefore, 3a = 180
If we shift 3 on the right side of the equation we will get,
$a=\dfrac{180}{3}$
Therefore, a = 60 cm - (3)
To proceed further in the solution we should know the formula given below,
Heron’s Formula:
Area of triangle $=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$ - (4)
Where, $s=\dfrac{Perimeter}{2}$ and a, b and c are the sides of the triangle,
Before calculating the area we should know the value of ‘s’ therefore,
$s=\dfrac{Perimeter}{2}$
If we put the value of equation (2) in the above equation we will get,
$\therefore s=\dfrac{180}{2}$
Therefore, s = 90 cm …………………………………………………… (5)
As we know that all the sides of equilateral triangle are equal and its value is a therefore the formula of equation (4) can be reduced to,
Area of triangle $=\sqrt{s\left( s-a \right)\left( s-a \right)\left( s-a \right)}$
Simplifying the above equation we will get,
Therefore, Area of triangle $=\sqrt{s{{\left( s-a \right)}^{2}}\left( s-a \right)}$
If we take the term ${{\left( s-a \right)}^{2}}$ out of the square root we will get,
Therefore, Area of triangle $=\left( s-a \right)\sqrt{s\left( s-a \right)}$
If we put the values of equation (3) and equation (5) in the above equation we will get,
Therefore, Area of triangle $=\left( 90-60 \right)\sqrt{90\left( 90-60 \right)}$
By simplifying the above equation we will get,
Therefore, Area of triangle $=\left( 30 \right)\sqrt{90\left( 30 \right)}$
By rearranging the above equation we will get,
Therefore, Area of triangle $=\left( 30 \right)\sqrt{900\left( 3 \right)}$
As we know that the square root of 900 is 30 therefore above equation will become,
Therefore, Area of triangle $=\left( 30 \right)30\sqrt{3}$
Further simplification in the above equation will give,
Therefore, Area of triangle $=900\sqrt{3}\,c{{m}^{2}}$
Therefore the area of the given equilateral triangle is $900\sqrt{3}\,c{{m}^{2}}$.

Note: In this question after finding the length of the side of the equilateral triangle we can cross check the answer by using the formula of the area of the equilateral triangle that $\dfrac{{\sqrt 3}{a}^2}{4}$.