IGCSE Mathematics: Similarity and Congruence Practice Questions
Similar shapes have the same angles and sides in the same ratio. If the length scale factor is k, the area scale factor is k squared and the volume scale factor is k cubed.
Topic 4.4 of Cambridge IGCSE Mathematics 0580 contains the single most commonly misapplied rule on the paper: using the length scale factor for an area or volume. The questions below force the squared and cubed versions into the working.
What you need to know for Similarity and Congruence
- Similar shapesCorresponding angles are equal and corresponding sides are in the same ratio. One shape is an enlargement of the other.
- Length scale factorDivide a length on the larger shape by the corresponding length on the smaller shape.
- Area scale factorThe square of the length scale factor. If lengths are in the ratio 3 to 5, areas are in the ratio 9 to 25.
- Volume scale factorThe cube of the length scale factor. Lengths in the ratio 3 to 5 give volumes in the ratio 27 to 125.
- Congruent shapesIdentical in both shape and size. Congruence can be proved by SSS, SAS, ASA or RHS.
- RHSRight angle, hypotenuse and one other side. It applies only to right-angled triangles.
IGCSE Mathematics Similarity and Congruence questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Two triangles are similar. In the smaller triangle a side of 6 cm corresponds to a side of 9 cm in the larger one. Another side of the smaller triangle is 8 cm. Calculate the corresponding side in the larger triangle.
Show the worked answer
- Find the length scale factor by dividing corresponding sides: 9 divided by 6 = 1.5.
- Every length in the larger triangle is 1.5 times the corresponding length in the smaller one.
- The unknown side = 8 x 1.5.
- The corresponding side is 12 cm.
Two similar solids have corresponding lengths in the ratio 3 to 5. The smaller solid has a volume of 54 cm3. Calculate the volume of the larger solid.
Show the worked answer
- The length scale factor from small to large is 5 divided by 3.
- The volume scale factor is the cube of that: (5 divided by 3)3 = 125 divided by 27.
- Larger volume = 54 x 125 divided by 27.
- 54 divided by 27 = 2, and 2 x 125 = 250 cm3.
Two similar rectangles have a length scale factor of 2.5. The smaller rectangle has an area of 20 cm2. Calculate the area of the larger rectangle.
Show the worked answer
- The area scale factor is the square of the length scale factor.
- 2.52 = 6.25.
- Larger area = 20 x 6.25.
- The area of the larger rectangle is 125 cm2.
State the four conditions that can be used to prove two triangles are congruent, and explain what congruent means.
Show the worked answer
- Congruent triangles are identical in both shape and size, so every pair of corresponding sides and angles is equal. One can be mapped onto the other by a reflection, rotation or translation.
- SSS: all three pairs of corresponding sides are equal.
- SAS: two pairs of sides and the angle between them are equal. The angle must be the included angle.
- ASA: two pairs of angles and a corresponding side are equal. RHS: both triangles have a right angle, and the hypotenuse and one other side are equal.
Common mistakes in this topic
- Using the length scale factor for area or volume.
- Treating similarity as adding a fixed amount to each side.
- Confusing similar with congruent. Congruent shapes are the same size, similar ones need not be.
- Using SSA as a congruence condition.
- Dividing the wrong way round when finding a scale factor.
Exam tips
- Write the scale factor as large divided by small if you are scaling up, and the reverse if scaling down.
- Before any area or volume calculation, write down whether you need k, k squared or k cubed.
- Sense check: the larger shape must give the larger answer.
- For congruence proofs, state the condition by name and list the three matching pairs.
- Identify corresponding sides by matching the angles, not by position on the page.
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Similarity and Congruence FAQs
What are the area and volume scale factors?
If the length scale factor is k, the area scale factor is k squared and the volume scale factor is k cubed. So doubling every length multiplies the area by four and the volume by eight. Applying k directly to an area or volume is the most common error in this topic.
What is the difference between similar and congruent?
Similar shapes have equal angles and sides in the same ratio, so one is an enlargement of the other and they can be different sizes. Congruent shapes are identical in both shape and size, so every corresponding side and angle is exactly equal.
How do I prove two triangles are congruent?
Use one of four conditions: SSS, three pairs of equal sides. SAS, two sides and the included angle. ASA, two angles and a corresponding side. RHS, a right angle with the hypotenuse and one other side. State the condition by name and list the three matching pairs.
Why is SSA not a valid congruence condition?
Two sides and an angle that is not between them can produce two different triangles, because the third vertex can fall in two positions. Since the triangle is not uniquely determined, SSA cannot prove congruence. The angle must be the included angle for SAS to work.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.