IGCSE Maths similarity and congruence: scale factors for length, area and volume

What each tier expects

The key ideas

Two shapes are similar if one is an enlargement of the other. For triangles, it is enough to show that the angles are equal, for example because of parallel lines (corresponding or alternate angles) or a shared angle. The classic exam diagram is a small triangle inside a large one, with a line parallel to the base.

The scale factor k is a ratio of corresponding lengths: large ÷ small. Label corresponding vertices in the same order so that you compare the right sides. Redrawing the two triangles separately, the same way round, makes this much easier.

Area scale factor = k^2 and volume scale factor = k^3. Working backwards, if you know an area ratio, take its square root to get k; from a volume ratio, take the cube root. This also applies to surface area and capacity, since capacity is a volume.

Congruent triangles are the same shape and size. The standard ways to recognise them are three equal sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS) and right angle, hypotenuse and side (RHS). Neither specification asks for formal congruence proofs, but recognising congruent shapes helps in angle and transformation questions.

Worked example 1: lengths in similar triangles

1. In triangle ABC, D is on AB and E is on AC with DE parallel to BC. AD = 4 cm, AB = 10 cm and DE = 6 cm. Find BC.
2. Triangles ADE and ABC are similar: angle A is shared and the angles at D and B are corresponding angles on parallel lines.
3. Scale factor k = AB ÷ AD = 10 ÷ 4 = 2.5.
4. BC = 2.5 × DE = 2.5 × 6 = 15 cm. Check: 15 ÷ 6 = 2.5, the same ratio as 10 ÷ 4.

Worked example 2: area and volume scale factors

1. Two similar bottles have heights 12 cm and 18 cm. The larger holds 810 ml. How much does the smaller hold?
2. Length scale factor from large to small: 12 ÷ 18 = 2/3.
3. Volume scale factor: (2/3)^3 = 8/27.
4. Smaller capacity = 810 × 8/27 = 240 ml. Check: 240 × 27/8 = 810.
5. If the label on the larger bottle has area 45 cm^2, the matching label on the smaller has area 45 × (2/3)^2 = 45 × 4/9 = 20 cm^2.

Worked example 3: working back from an area ratio

1. Two similar shapes have areas 20 cm^2 and 45 cm^2. A side of the smaller is 6 cm. Find the matching side of the larger.
2. Area ratio = 45 ÷ 20 = 2.25.
3. Length scale factor = sqrt(2.25) = 1.5.
4. Matching side = 6 × 1.5 = 9 cm. Check: (9 ÷ 6)^2 = 2.25. Correct.

Common mistakes that cost marks

• Using the length scale factor directly for areas or volumes.
• Matching the wrong sides because the triangles face different ways; redraw them.
• In the parallel-line diagram, using AD and DB instead of AD and AB for the ratio.
• Subtracting lengths to find a scale factor instead of dividing.
• Assuming shapes are similar because they look alike: the angles or ratios must be shown.
• Forgetting that capacity in litres or ml is a volume, so it scales by k^3.

Exam technique and how a tutor helps

Write the scale factor as a clear line, such as 'k = 18 ÷ 12 = 1.5', and state which way it goes. When a question mixes lengths, areas and volumes, write all three scale factors at the start: k, k^2 and k^3. For Cambridge Extended 'show that the triangles are similar' parts, give each pair of equal angles with its reason.

Similarity often hides inside other topics: shadows and heights, maps and scale drawings, vectors with ratios, and frustums of cones. Spotting it is half the question.

A one-to-one tutor helps most at the spotting stage. On the shared whiteboard the tutor can pull the two triangles apart and redraw them side by side, which is the step most students skip. Practice then moves to area and volume questions, where the main habit to build is asking 'is this a length, an area or a volume?' before touching the numbers.

Self-check: can you do these?

• Lengths are multiplied by 3. What happens to areas and volumes? (Answer: × 9 and × 27)
• Two similar cones have volumes 54 cm^3 and 250 cm^3. Find the ratio of their heights. (Answer: 3 : 5)
• A map has scale 1 : 50 000. What real distance is 4 cm on the map? (Answer: 2 km)
• Similar triangles have corresponding sides 8 cm and 12 cm. The smaller has area 32 cm^2. Find the larger area. (Answer: 72 cm^2)
• State two facts about corresponding angles and sides in similar shapes.