Areas of similar shapes || IGCSE, O level Math B - Mathematics Gurukul, GOLN

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Areas of similar shapes

The formula for the area of similar shapes is given below:

$\frac{A r e a o f f i g u r e A}{A r e a o f f i g u r e B} = {(\frac{a}{b})}^{2}$

To find the area of two similar shapes we can use the knowledge that the ratio of their areas is equal to the ratio of the square of their respective sides.

Let us take an example; △ABC $\sim$ △DEF

So their corresponding parts must be proportional

$\frac{A B}{D E} = \frac{A C}{D F} = \frac{B C}{E F}$

$\frac{A r e a o f △ A B C}{A r e a o f △ D E F} = {(\frac{A B}{D E})}^{2} = {(\frac{A C}{D F})}^{2} = {(\frac{B C}{E F})}^{2}$

Two squares are similar. In the figure below, let the sides of square 1 and 2 be s1 and s2 respectively.

Example 1: If the corresponding sides of two similar triangles are in the ratio 4:5, what is the ratio of their areas?

Solution:

We know that in similar figures the ratio of areas is equal to the ratio of the squares of their respective sides.

Let the ratio of sides of two triangles be $\frac{s_{1}}{s_{2}} = \frac{4}{5}$

$\frac{A r e a o f t r i a n g l e 1}{A r e a o f t r i a n g l e 2} = {(\frac{s_{1}}{s_{2}})}^{2}$

$= {(\frac{4}{5})}^{2}$

$= \frac{16}{25}$

Therefore, the ratio of area of two triangles is 16 : 25.

Example 2: Find the area of a smaller figure.

Solution:

Since the shapes of both the figures are the same, they are similar. We know that in similar figures the ratio of their areas is equal to the ratio of the squares of their respective sides.

$\frac{A r e a o f b i g g e r f i g u r e}{A r e a o f s m a l l e r f i g u r e} = {(\frac{S i d e o f b i g g e r f i g u r e}{S i d e o f s m a l l e r f i g u r e})}^{2}$

⇒ $(\frac{196}{A r e a o f s m a l l e r f i g u r e}) = {(\frac{7}{9})}^{2}$ [Substitute the given values]

⇒ $(\frac{196}{A r e a o f s m a l l e r f i g u r e}) = (\frac{49}{25})$

⇒ Area of smaller figure × 49 = 196 × 25 [Cross multiplication]

⇒ Area of smaller figure = $\frac{196 \times 25}{49}$ [Simplify]

∴ Area of smaller figure = $100 m^{2}$

Therefore, the area of smaller figure is $100 m^{2}$

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Areas of similar shapes

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