Congruence and similarity

3) Lengths, areas and volumes of similar shapes

Ratio of areas

The lengths of the larger square are 3 times the lengths of the smaller square. The scale factor is 3. The area of the smaller square is 4 cm2. The area of the larger square is 36 cm2.

The area of the larger square is 9 times larger than the area of the smaller square.

The ratio of areas is $1:9$. This is $1:3^{2}$.

If the scale factor is $k$, the ratio of areas is $k^{2}$.

Example 1

These two clocks are similar. The area of the small clock face is approximately 28.3 cm2. Calculate the area of the face of the larger clock.

The scale factor = $\frac{24}{6}=4$
The ratio of areas is = $1:4^{2}=1:16$
$larger \: area = 16 \times smaller \: area$
$28.3 \times 16 = 452.8$

The area of the large clock face is approximately 452.8 cm2.

Example 2

These two pieces of paper are similar. The area of an A3 piece of paper is double the area of an A4 piece of paper. Calculate the width of the smaller piece of paper.

The ratio of areas is $1:2$. The ratio of lengths is $1:\sqrt{2}$
In this case, the scale factor is $\sqrt{2}$.
$smaller \: length = larger \: length \div \sqrt{2}$
$29.7 \div \sqrt{2} = 21 \: cm$

Ratio of volumes

The lengths of the larger square are 4 times the lengths of the smaller square.The scale factor is 4. The volume of the smaller cube is 8 cm3. The volume of the larger cube is 512 cm3.

The ratio of volumes is $8:512=1:64$. This is $1:4^{3}$

If the scale factor is $k$, the ratio of volumes $k^{3}$.

Example

These two tins of soup are similar. Calculate the diameter of the larger tin of soup.

The ratio of volumes is 125 : 500 = 1 : 4
The ratio of lengths is $1:\sqrt[3]{4}$
The scale factor is: $\sqrt[3]{4}$
$larger \: diameter = smaller \: diameter \times \sqrt[3]{4}$
$5 \times \sqrt[3]{4} = 7.9 \: cm \: (1 \: d.p.)$

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