Lengths, areas and volumes of similar shapes - Higher
Area scale factor
The lengths of the larger square are three times the length of the smaller square.
The length The ratio of corresponding lengths in similar shapes, ie how much larger or smaller the shapes are. is 3.
The area of the smaller square is 9 cm2. The area of the larger square is 81 cm2.
The area of the larger square is nine times as large as the area of the smaller square.
The area scale factor is 9. This is the length scale factor squared.
If the length scale factor is \(k\), the area scale factor is \(k^2\).
Example
These 2 clocks are similar. The area of the small clock face is approximately 50.3 cm2. Calculate the area of the face of the larger clock.
The length scale factor = \(\frac{24}{6} = 4\)
The area scale factor = \(4^2 = 16\)
\(\text{Larger area} = 16 \times \text{smaller area}\)
\(50.3 \times 16 = 804.8\)
The area of the large clock face is approximately 804.8 cm2.
Question
These 2 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.
Area scale factor is 2.
Length scale factor is: \(\sqrt{\text{area scale factor}}\)
In this case, the length scale factor is \(\sqrt{2}\). Do not round this number yet.
\(\text{Smaller length} = \text{larger length} \div \sqrt{2}\)
\(29.7 \div \sqrt{2} = 21~\text{cm}\)
Volume scale factor
The lengths of the larger square are four times the length of the smaller square.
The length scale factor is 4.
The volume of the smaller cube is 8 cm3. The volume of the larger cube is 512 cm3.
The volume scale factor is 64. This is the length scale factor cubed.
If the length scale factor is \(k\), the volume scale factor is \(k^3\).
Example
These 2 tins of soup are similar. Calculate the diameter of the larger tin of soup.
The volume scale factor is 4.
Length scale factor is: \(\sqrt[3]{\text{volume scale factor}}\)
In this case, the length scale factor is \(\sqrt[3]{4}\). Do not round this number yet.
\(\text{Larger diameter} = \text{smaller diameter} \times \sqrt[3]{4}\)
\(5 \times \sqrt[3]{4} = 7.9~\text{cm}\) (1 dp)
