Similarity and congruence
Similar shapes, scale factors for area and volume, and congruence conditions.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 4.3.1Use similarity to find lengths, and apply area and volume scale factors.
- 4.3.2Identify congruent triangles using the standard conditions.Supplement
6 minute read
Similarity and congruence
Similar shapes
Two shapes are similar if their angles are equal and their corresponding sides are in the same ratio. Same shape, different size. The linear scale factor k is found by dividing a length on the new shape by the corresponding length on the original.
The scale factor rule — the key idea
- Length scale factor =
k - Area scale factor =
k² - Volume scale factor =
k³
This follows from dimensions: area is two lengths multiplied, volume is three. So if a model is 3 times as long, its surface area is 9 times as large and its volume 27 times as large. Assuming volume triples is the single most common error in the topic. Working backwards: if the volume is 8 times larger, then k³ = 8, so k = 2 and lengths are doubled.
Congruence
Congruent shapes are identical in both shape and size — one can be placed exactly on the other, allowing rotation or reflection. The four conditions for congruent triangles:
- SSS — three sides equal.
- SAS — two sides and the included angle equal.
- ASA (or AAS) — two angles and a corresponding side.
- RHS — right angle, hypotenuse and one other side.
Note SSA is not a condition — two sides and a non-included angle can produce two different triangles.
Similar versus congruent
Congruent shapes are similar with a scale factor of exactly 1. All congruent shapes are similar; almost no similar shapes are congruent.
Think of it like this
Doubling every length of a box gives four times the wrapping paper and eight times the contents. The powers are not a rule to memorise — they are just how many lengths get multiplied together.
Worked examples
Method, step by step
Two similar cones have heights 4 cm and 10 cm. The smaller has volume 32 cm³. Find the volume of the larger.
- 1Linear scale factor k = 10/4 = 2.5.
- 2Volume scale factor = k³ = 2.5³ = 15.625.
- 3Larger volume = 32 × 15.625 = 500 cm³.
500 cm³
Two similar shapes have areas 18 cm² and 50 cm². Find the ratio of their corresponding lengths.
- 1Area scale factor = 50/18 = 25/9.
- 2Length scale factor = √(25/9) = 5/3.
- 3So corresponding lengths are in the ratio 3 : 5.
3 : 5
Common misconceptions
- Using the linear scale factor for area or volume, e.g. thinking a shape twice as long has twice the area rather than four times.
- Treating SSA as a congruence condition — it is not, because two distinct triangles can satisfy it.
- Comparing non-corresponding sides when finding a scale factor. Match sides by their position between equal angles.
In the exam
- Write `k`, `k²` and `k³` at the top of the working and decide which is needed before calculating.
- For congruence proofs, name the condition (SSS, SAS, ASA, RHS) explicitly — that name is a mark on its own.