Lengths, areas and volumes of similar shapes - Higher
Ratio of areas
The lengths of the larger square are three times the lengths of the smaller square.
The 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 4 cm2. The area of the larger square is 36 cm2.
The area of the larger square is nine times as large as 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
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 = \(1 : 4^2 = 1 : 16\)
\(\text{Larger area} = 16 \times \text{smaller area}\)
28.3 x 16 = 452.8
The area of the large clock face is approximately 452.8 cm2.
Question
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 \(\sqrt{2}\).
\(\text{Smaller length} = \text{larger length} \div \sqrt{2}\)
\(29.7 \div \sqrt{2} = 21~\text{cm}\)
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 is \(k^3\).
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: \(\sqrt[3]{4}\)
\(\text{Larger diameter} = \text{smaller diameter} \times \sqrt[3]{4}\)
\(5 \times \sqrt[3]{4} = 7.9~\text{cm}\) (1 dp)
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- Angles, lines and polygons - AQA
- Loci and constructions - AQA
- 2-dimensional shapes - AQA
- 3-dimensional shapes - AQA
- Circles, sectors and arcs - AQA
- Circle theorems - Higher - AQA
- Pythagoras' theorem - AQA
- Units of measure - AQA
- Trigonometry - AQA
- Vectors - AQA
