Question

A traffic signal board indicates ‘SCHOOL AHEAD’ is an equilateral triangle with side ‘a’. Find the area of the traffic board using Heron’s formula. If its perimeter is ${\text{180cm}}$ . What will be the area of the signal board?

Answer 1

Hint:In this question they have given that the traffic board is in the shape of equilateral triangle. By using Heron's formula, the area of the equilateral triangle can be calculated. The semi-perimeter is the half of the perimeter.

Formula used:The formula used for this question is,
The area of the equilateral triangle by using Heron’s formula is,
 ${\text{Area of the equilateral triangle}} = \sqrt {{\text{s(s}} - {\text{a)(s}} - {\text{b)(s}} - {\text{c)}}} $
Where,
 ${\text{s}}$ be the semi-perimeter
 ${\text{a,b,c}}$ be the sides of the equilateral triangle

Complete step-by-step answer:
The data given in the question are,
The traffic board is in the shape of an equilateral triangle.
The perimeter of the traffic board is ${\text{180cm}}$
The side of the equilateral triangle is ${\text{a}}$

The semi-perimeter of the traffic board is,
 ${\text{s}} = \dfrac{{{\text{perimeter}}}}{2}$
Substitute the value of perimeter in the above equation we get,
 ${\text{s}} = \dfrac{{180}}{2}$
The value of semi-perimeter of the traffic board is,
 ${\text{s}} = 90{\text{cm}}$
From the given, the shape of the traffic board is in equilateral triangle. In equilateral triangle all the three sides of the triangle are equal.
Then, all the sides of the traffic board be ${\text{a}}$
 ${\text{perimeter}} = {\text{a + a + a}}$
By solving we get,
 ${\text{perimeter}} = 3{\text{a}}$
While substituting the perimeter value we will get,
 $180 = 3{\text{a}}$
Then the side of the triangle will be,
 \[{\text{a}} = \dfrac{{180}}{3}\]
By dividing we get,
 ${\text{a}} = 60{\text{cm}}$
The sides of the equilateral triangle are $60{\text{cm}}$
Then,
 ${\text{Area of the traffic board}} = \sqrt {{\text{s(s}} - {\text{a)(s}} - {\text{a)(s}} - {\text{a)}}} $
By substituting all the values in the above equation we get,
 $ \Rightarrow \sqrt {{\text{90(90}} - 60{\text{)(90}} - 60{\text{)(90}} - 60{\text{)}}} $
By solving we get,
 $ \Rightarrow \sqrt {{\text{90(3}}0{\text{)(30)(30)}}} $
 $ \Rightarrow \sqrt {{{30}^2} \times {{30}^2} \times 3} $
By cancelling the square and square root we get,
 $ \Rightarrow 30 \times 30\sqrt {\text{3}} $
By multiplying we get,
 \[{\text{Area of the traffic board}} = 900\sqrt 3 {\text{c}}{{\text{m}}^2}\]

 $\therefore $The area of traffic signal board is ${\text{900}}\sqrt 3 {\text{c}}{{\text{m}}^2}$
Hence, the area of the traffic signal board is in ${\text{900}}\sqrt 3 {\text{c}}{{\text{m}}^2}$ .

Note:The perimeter of any shape is the sum of all the sides of that shape. The unit of the measurement is the most important thing in these types of questions. And the Semi-perimeter has to be calculated with more attention, it may lead to incorrect solutions.