Name: Area of Quadrilateral Using Heron's Formula
Uploaded: 2019-07-19
Duration: 10 min 24 s
Description: Did you know that the area of a quadrilateral can be calculated by dividing the quadrilateral into two triangles and using Heron's formula? Let's watch this...

Question

A traffic signal board, indicating

‘ S C H O O L A H E A D ’

, is an equilateral triangle with side

‘ a ’ .

Find the area of the signal board, using Heron’s formula. If its perimeter is

180 c m,

what will be the area of the signal board?

Accepted Answer

Given,

The length of one side of a traffic signal board $= a c m$

We know that the perimeter of an equilateral triangle $= 3 \times side$

Hence, the perimeter of the traffic signal board $= 3 a c m$

And, the semi-perimeter of the traffic signal board, $s = \frac{3 a}{2} c m$

We know that according to Heron’s Formula,

Area of the triangle $= \sqrt{s (s - a) (s - b) (s - c)}$ ,

where $s$ is the semi-perimeter and $a, b, c$ are the length of the sides.

So, the area of traffic signal board is,

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

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

$= \frac{\sqrt{3} a^{2}}{4} c m^{2}$

Given,

The perimeter of the signal board $= 180 c m$

$\Rightarrow 180 = 3 a$

$\Rightarrow a = 60 c m$

Hence, the area of traffic signal board is,

$= \frac{\sqrt{3} a^{2}}{4}$

$= \frac{\sqrt{3}}{4} \times 60^{2} {cm}^{2}$

$= \sqrt{3} \times 900 c m^{2}$

$= 900 \sqrt{3} c m^{2}$

Thus, the area of the signal board is $900 \sqrt{3} c m^{2}$

EXERCISE

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