Question

Find the area of the pentagon shown in the fig, if $AD = 8 cm$ , $AH=6 cm$ , $AG = 4 cm$ , $AF = 3 cm$ , $BF = 2 cm$ , $CH = 3 cm$ and $EG = 2.5 cm$ .

Accepted Answer

Given that in pentagon $A B C D E$ , $EG ⊥ AD, EF ⊥ AD & CH ⊥ AD$ .

$CH = 3 cm$ and $EG = 2.5 cm$ .

We are asked to find the area of the pentagon.

We can see that the pentagon is divided into different triangles and trapezium. So the area of the pentagon will be equal to the sum of the areas of these figures.

That is,

Area of pentagon $A B C D E =$ Area of $△ A B F +$ Area of trapezium $BCFH +$ Area of $△ CHD +$ Area of $△ ADE$

Consider $△ ABF$

Area of a triangle $= \frac{1}{2} \times base \times height$

So, the area of $△ ABF = \frac{1}{2} \times AF \times BF$

$= \frac{1}{2} \times 3 cm \times 2 cm$

$= \frac{1}{2} \times 3 \times 2 {cm}^{2}$

$= 3 {cm}^{2}$

Similarly, area of $△ C H D = \frac{1}{2} \times C H \times D H$

$AD - AH = DH \Rightarrow 8 cm - 6 cm \Rightarrow DH = 2 cm$

$∴$ Area of $△ CHD = \frac{1}{2} \times 3 cm \times 2 cm$

$= \frac{1}{2} \times 3 \times 2 {cm}^{2}$

$= 3 {cm}^{2}$

Now, area of $△ ADE = \frac{1}{2} \times AD \times GE$

$= \frac{1}{2} \times 8 cm \times 2.5 cm$

$= 4 \times 2.5 {cm}^{2}$

$= 10 {cm}^{2}$

Now, the area of the trapezium $= \frac{1}{2} \times (Sum of parellel sides) \times (height)$

So, the area of trapezium $BCFH = \frac{1}{2} \times (BF + CH) \times FH$

$FH = AH - AF = 6 cm - 3 cm = 3 cm$

So the area of trapezium $= \frac{1}{2} \times (2 cm + 3 cm) \times 3 cm$

$= \frac{1}{2} \times 5 cm \times 3 cm$

$= \frac{15}{2} {cm}^{2}$

$= 7.5 {cm}^{2}$

Now, area of the pentagon $A B C D E = area (△ A B F) + area (trapezium B C F H) + area (△ C H D) + area (△ A D E)$

$= 3 {cm}^{2} + 7.5 {cm}^{2} + 3 {cm}^{2} + 10 {cm}^{2}$

$= (3 + 7.5 + 3 + 10) {cm}^{2}$

$= 23.5 {cm}^{2}$

Hence, the area of the pentagon $A B C D E$ is $23.5 {cm}^{2}$ .

S Purkayastha Solutions for Chapter: Area of Polygons, Exercise 3: CHECK POINT

S Purkayastha Mathematics Solutions for Exercise - S Purkayastha Solutions for Chapter: Area of Polygons, Exercise 3: CHECK POINT

Questions from S Purkayastha Solutions for Chapter: Area of Polygons, Exercise 3: CHECK POINT with Hints & Solutions

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