AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Compare lengths, areas and volumes using ratio notation scale factors
Full specification wording
Ratio, proportion and rates of change: compare lengths, areas and volumes using ratio notation scale factors
Similar shapes have equal corresponding angles and corresponding lengths in a fixed ratio.
Checked against the AQA 8300 specification (opens in a new tab) on .
- Point 14 of 21
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Similar shapes have equal corresponding angles and corresponding lengths in a fixed ratio. If the length scale factor is $k$, then every length is multiplied by $k$, every area by $k^2$, and every volume by $k^3$. For example, doubling lengths ($k = 2$) multiplies areas by $2^2 = 4$ and volumes by $2^3 = 8$. Ratio notation can compare lengths, areas or volumes directly: a length ratio $1 : 3$ gives an area ratio $1^2 : 3^2 = 1 : 9$ and a volume ratio $1^3 : 3^3 = 1 : 27$. To compare, identify whether the given measures are lengths, areas or volumes, find the relevant scale factor, and apply the correct power. To find a length scale factor from an area ratio, take the square root; from a volume ratio, take the cube root.
What earns the marks
- Identifies corresponding lengths, areas or volumes in similar figures.
- Uses the length scale factor $k$ and applies $k^2$ to areas and $k^3$ to volumes.
- Writes ratios in the correct order and simplifies where appropriate.
- Finds a length scale factor by square-rooting an area ratio or cube-rooting a volume ratio.
- Uses ratio notation to compare two similar figures, for example length ratio $1 : 3$, area ratio $1 : 9$, volume ratio $1 : 27$.
- Checks that the power used matches the dimension being compared.
Common mistakes and how to put them right
Applying the length scale factor directly to an area or volume.
Put it right: Square it for area and cube it for volume.
Confusing area and volume ratios, such as treating $1.
Put it right: Match the power to the dimension.
Forgetting to square-root or cube-root when working back from an area or volume ratio to a length ratio.
Put it right: Use the inverse power.
Reversing the order of a ratio.
Put it right: Keep the order consistent with the figures being compared.
Exam tips
- Label each given measure as length, area or volume before choosing a scale factor.
- Write the length scale factor first, then derive area and volume factors from it.
- Use ratio notation to show the comparison clearly, for example $1 : 3$ for lengths.
- Check whether the question asks for a scale factor or a ratio, and answer in that form.
Try the lesson
The start of the Ratio, proportion and rates of change: compare lengths, areas and volumes using ratio notation scale factors lesson, made from checked teaching notes for this specification point.
Have a go first
Two mathematically similar solid shapes have corresponding lengths in the ratio 1 : 3. What is the ratio of their surface areas?
Listen
Similar shapes have equal corresponding angles and corresponding lengths in a fixed ratio. If the length scale factor is k, then every length is multiplied by k, every area by k^2, and every volume by k^3. For example, doubling lengths (k = 2) multiplies areas by 2^2 = 4 and volumes by 2^3 = 8. Ratio notation can compare lengths, areas or volumes directly: a length ratio 1 : 3 gives an area ratio 1^2 : 3^2 = 1 : 9 and a volume ratio 1^3 : 3^3 = 1 : 27. To compare, identify whether the given measures are lengths, areas or volumes, find the relevant scale factor, and apply the correct power. To find a length scale factor from an area ratio, take the square root; from a volume ratio, take the cube root. A strong answer earns marks for these points. Identifies corresponding lengths, areas or volumes in similar figures. Uses the length scale factor k and applies k^2 to areas and k^3 to volumes. Writes ratios in the correct order and simplifies where appropriate. Finds a length scale factor by square-rooting an area ratio or cube-rooting a volume ratio. A common slip: Applying the length scale factor directly to an area or volume. Instead, square it for area and cube it for volume. Exam tip: Label each given measure as length, area or volume before choosing a scale factor.
Watch it explained
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Similar shapes have equal corresponding angles, and every corresponding length is in the same fixed ratio. If the length scale factor is k, every length is multiplied by k. This gives the starting point for comparing lengths, areas and volumes. The length scale factor controls area and volume through powers. If lengths scale by k, every area scales by k squared, and every volume scales by k cubed. For example, doubling lengths means k equals two, so areas multiply by four and volumes by eight. To compare similar figures, first identify whether each given measure is a length, an area or a volume. Then find the relevant scale factor and apply the correct power. Check that the power matches the dimension being compared. A length ratio is one to three. Pause and think: what area ratio and volume ratio follow? The length scale factor is three. The area ratio is one squared to three squared, which is one to nine. The volume ratio is one cubed to three cubed, which is one to twenty-seven. A common mistake is applying the length scale factor directly to an area or volume. For area, square the length factor; for volume, cube it. Another mistake is confusing area and volume ratios, such as treating one to nine as a volume ratio when it is an area ratio. Keep the ratio order consistent with the figures being compared. Sometimes you know an area ratio or a volume ratio and need a length ratio. From an area ratio, take the square root. From a volume ratio, take the cube root. This uses the inverse power and returns the length scale factor. Remember: similar shapes use one length scale factor, k. Lengths scale by k, areas by k squared, and volumes by k cubed. Ratio notation can show these comparisons directly, such as one to three for lengths, one to nine for areas, and one to twenty-seven for volumes. Check the dimension, then choose the matching power.
Key words
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Similarity and scale factors
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Length, area and volume relationships
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Ratio notation for comparison
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Powers and roots in scale-factor problems
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See every definition
- Similarity and scale factors
- Similar shapes have equal corresponding angles and corresponding lengths in a fixed ratio, and if the length scale factor is k then every length is multiplied by k.
- Length, area and volume relationships
- If the length scale factor is k, then every length is multiplied by k, every area by k squared, and every volume by k cubed.
- Ratio notation for comparison
- Ratio notation can compare lengths, areas or volumes directly, so a length ratio 1 : 3 gives an area ratio 1 : 9 and a volume ratio 1 : 27.
- Powers and roots in scale-factor problems
- To find a length scale factor from an area ratio, take the square root; from a volume ratio, take the cube root.
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