Question

A traffic board signal indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board using Heron’s formula. It is given that the perimeter of the triangle is 180 cm.
$$\mathrm A.\;900\;\mathrm{cm}^2\\\mathrm B.\;900\sqrt2\;\mathrm{cm}^2\\\mathrm C.\;900\sqrt3\;\mathrm{cm}^2\\\mathrm D.\;2700\;\mathrm{cm}^2$$

Answer 1

Hint: To solve this problem, Heron’s formula for the area of a triangle will be used. If the sides of the triangle are a, b and c and s is the semi-perimeter of the triangle then-
$$\mathrm{Area}=\sqrt{\mathrm s\left(\mathrm s-\mathrm a\right)\left(\mathrm s-\mathrm b\right)\left(\mathrm s-\mathrm c\right)}$$

Complete step-by-step solution -
We have been given that the perimeter of the equilateral triangle is 180 cm. The side of the equilateral triangle is ‘a’. Then we can write that-
P = a + a + a = 180
3a = 180
a = 60 cm
Here S =$\dfrac{P}{2} $ = $\dfrac{180}{2}$
Hence, the sides of the equilateral triangle are 60 cm each.
Using Heron’s formula we can calculate the area as-
$$\mathrm A=\sqrt{\dfrac{180}2\left(\dfrac{180}2-60\right)\left(\dfrac{180}2-60\right)\left(\dfrac{180}2-60\right)}\\\mathrm A=\sqrt{90\left(30\right)\left(30\right)\left(30\right)}\\\mathrm A=\sqrt{3\times30\times30\times30\times30}\\\mathrm A=30\times30\sqrt3\\\mathrm A=900\sqrt3\;\mathrm{cm}^2$$

This is the answer. Hence, the correct option is C.

Note: In such types of problems, we may not be given the value of a, b, c and s directly. We have to first find these values using the given information and then solve. Also, when writing the final answer, do not forget to write the units. If in this question it is not asked about using the heron’s formula we could use the area of triangle as $\dfrac{1}{2} \times \text{base} \times \text{height} $