Congruence & SimilarityCambridge IGCSE Maths: Revision notes

Section 1

What does it mean for shapes to be congruent?

Two shapes are congruent if they are identical in both shape and size — one could be placed exactly on top of the other (possibly after a rotation, reflection or translation). All corresponding sides and angles are equal.

At this level, candidates are not expected to formally prove congruence — you just need to recognise and use congruent shapes' equal properties.

Key termscongruent

Section 2

What does it mean for shapes to be similar?

Two shapes are similar if they have the same shape but not necessarily the same size — all corresponding angles are equal, and corresponding sides are in the same ratio (the scale factor).

To show two triangles are similar (Extended): show that their corresponding angles are equal (AA), or that their corresponding sides are all in the same ratio (SSS), or that two sides are in the same ratio with the included angle equal (SAS).

Key termssimilarscale factor

Exam tip

When explaining similarity, state which angles are equal and why (e.g. 'corresponding angles are equal because the lines are parallel').

Section 3

How do I calculate lengths in similar shapes?

Identify a pair of corresponding sides where both lengths are known — divide to find the scale factor
Multiply the known length in the other shape by the scale factor to find the unknown length

Example: Triangle ABC is similar to triangle PQR. AB = 4 cm corresponds to PQ = 10 cm, so the scale factor is 10 ÷ 4 = 2.5. If BC = 6 cm, then QR = 6 × 2.5 = 15 cm.

Key termscorresponding sides

Example

A photo 8 cm wide is enlarged to a similar poster 24 cm wide (scale factor 3). If the photo is 5 cm tall, the poster is 5 × 3 = 15 cm tall.

Section 4

How do length, area and volume ratios relate for similar shapes? (Extended)

If two similar shapes have a length ratio of $k$ , then:

Ratio type	Relationship
Length ratio	$k$
Area ratio	$k^2$
Volume ratio	$k^3$

This is because area scales with two dimensions and volume scales with three.

Key termslength ratioarea ratiovolume ratio

Example

Two similar cylinders have a length (height) ratio of 3:1. Their volume ratio is 3³:1³ = 27:1.

Section 5

How do I solve problems involving similarity ratios?

Identify the length scale factor $k$ from given corresponding lengths
For an area problem, square $k$ ; for a volume problem, cube $k$
Multiply (or divide) the known area/volume by this squared/cubed factor to find the unknown

Example: Two similar containers have heights 4 cm and 12 cm (length ratio 1:3). If the smaller holds 50 cm³, the larger holds $50 \times 3^3 = 1350$ cm³.

Common mistake

Applying the length scale factor directly to an area or volume (instead of squaring or cubing it first) is one of the most common exam errors on this topic.

Must Know

Congruent shapes are identical in shape and size; similar shapes have equal angles and proportional sides
Scale factor = ratio of corresponding sides between similar shapes
To find an unknown length: find the scale factor from a known pair, then multiply
If length ratio is k, area ratio is k² and volume ratio is k³
Never apply the length scale factor directly to areas or volumes — square or cube it first
Formal proof of congruence is not required, but you must be able to use it

That's the notes covered.

Carry on to the next subtopic.