Comparing dimensions and similarity ratios
Edexcel 1MA1 tests this on both tiers but the most demanding versions sit on Higher Paper 2H or 3H. The skill links ratio with similar shapes and with area / volume scaling.
Length scale factor (k)
If two shapes are similar with linear scale factor k, then every corresponding length is multiplied by k.
Area scale factor
Area scales by k². Two similar triangles with sides in ratio 1 : 3 have areas in ratio 1 : 9.
Volume scale factor
Volume scales by k³. Two similar cones with heights in ratio 2 : 5 have volumes in ratio 8 : 125.
Working backwards
If you are given areas in ratio a : b, the linear ratio is √a : √b. From volumes in ratio a : b, the linear ratio is ∛a : ∛b.
✦Worked example
Two similar bottles. Smaller has volume 250 ml, larger has volume 2 litres = 2000 ml.
- Volume ratio = 250 : 2000 = 1 : 8.
- Linear scale factor = ∛8 = 2.
- If smaller bottle is 12 cm tall, larger is 12 × 2 = 24 cm.
Common Edexcel mark-scheme phrasing
- M1 for setting up a correct ratio.
- M1 for taking the appropriate root (square / cube).
- A1 for the correct final length / area / volume.
- B1 for stating the scale factor explicitly.
⚠Common mistakes— Common errors
- Forgetting to square or cube — applying the linear scale factor to area or volume directly.
- Taking √ when ∛ was needed (or vice versa).
- Mixing units (ml vs litres, cm² vs m²) without converting first.
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