How do you use similar shapes to find missing lengths, areas and volumes?
Using similarity to find missing lengths in similar shapes, and applying the linear, area and volume scale factors between similar figures.
A focused answer to the SQA National 5 Mathematics similarity content, covering similar shapes and the linear scale factor, finding missing lengths in similar triangles, and the area and volume scale factors as the square and cube of the linear scale factor.
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What this dot point is asking
The SQA wants you to recognise similar shapes, use the linear scale factor to find missing lengths in similar figures (especially similar triangles), and apply the area scale factor (the square of the linear scale factor) and the volume scale factor (the cube) to find missing areas and volumes.
Similar shapes and the linear scale factor
Similar shapes have the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio. That common ratio is the linear scale factor.
Similar triangles
Triangles are similar if their angles match (for example, when a line is drawn parallel to one side, or in nested triangles sharing an angle). Set up the ratio of corresponding sides and solve.
Area and volume scale factors
When you enlarge a shape, area grows faster than length, and volume faster still, because area depends on two dimensions and volume on three.
These factors also work in reverse. If you are told the areas and asked for the linear scale factor, take the square root of the area scale factor; for volumes, take the cube root.
Examples in context
Scale factors explain why size changes have outsized effects. A model car at scale has of the surface area to paint and of the volume of material. In cooking, doubling every length of a tin multiplies its capacity by , so the recipe must be scaled accordingly. Map and plan reading uses the linear scale factor directly to convert between drawing and real distances.
Try this
Q1. Two similar triangles have bases cm and cm. Find the scale factor. [1 mark]
- Cue. .
Q2. Two similar shapes have lengths in ratio . The smaller has area cm. Find the larger area. [2 marks]
- Cue. cm.
Q3. Two similar solids have lengths in ratio . The smaller has volume cm. Find the larger volume. [2 marks]
- Cue. cm.
Exam-style practice questions
Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
SQA National 5 20183 marksTwo triangles are similar. The smaller has a base of cm; the larger has a corresponding base of cm. If a side of the smaller is cm, find the corresponding side of the larger.Show worked answer →
Find the linear scale factor from the bases: (1 mark). Multiply the known side by the scale factor: cm (2 marks). Markers reward the scale factor and the enlarged side. Always divide the larger by the smaller to enlarge.
SQA National 5 20223 marksTwo similar bottles have heights cm and cm. The smaller holds ml. Calculate the volume of the larger bottle.Show worked answer →
The linear scale factor is (1 mark). Volume scales by the cube of the linear scale factor: (1 mark). Multiply: ml (1 mark). Markers reward cubing the linear scale factor and applying it to the volume.
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Sources & how we know this
- SQA National 5 Mathematics Course Specification — SQA (2018)