Geometry · GCSE Maths
Congruence and similarity
GCSE Maths congruence and similarity: congruent shapes are identical in size and shape; similar shapes have equal angles and sides in proportion — scale factor links matching lengths.
Congruent means same size and shape — one could sit on the other. Similar means same angles, sides in proportion. Scale factor multiplies every matching length.
The important bits
What you need to know
- 1
Congruent shapes have equal corresponding sides and equal corresponding angles. They are identical in size and shape.
- 2
Similar shapes have equal corresponding angles and sides in proportion. One is an enlargement or reduction of the other.
- 3
Congruence tests for triangles: SSS, SAS, ASA, RHS. If one test holds, the triangles are congruent.
- 4
For similar triangles, matching sides are in the same ratio. If the scale factor is 3, every length on the larger triangle is 3 times the corresponding length on the smaller.
- 5
Scale factor from small to large: k = large side ÷ small corresponding side. From large to small: k = small ÷ large.
- 6
Area scale factor is k². Volume scale factor is k³. If lengths double, area quadruples and volume multiplies by 8.
- 7
Parallel lines create similar triangles: a line parallel to one side of a triangle makes a smaller similar triangle on top.
- 8
Show congruence or similarity in proofs by stating equal angles (alternate, corresponding, vertically opposite) and equal or proportional sides, with reasons.
Quotations worth analysing
Short evidence. Real method.
“Corresponding sides are in the same ratio”
You must pair sides correctly: shortest with shortest, not shortest with longest. Mis-pairing gives a scale factor that fits no whole diagram.
“Scale factor k: new length = k × old length”
k > 1 enlarges; 0 < k < 1 reduces. A negative k reflects as well as scales — less common on Foundation but valid on enlargement grids.
“Area factor = k², volume factor = k³”
Doubling lengths quadruples area. Students who double area when lengths double invent a scale factor of 2 on area, which is wrong.
Go deeper
Congruence tests you can quote
SSS: three pairs of equal sides. SAS: two pairs of equal sides with the included angle equal. ASA: two angles and the included side. RHS: right angle, hypotenuse and one other side equal — the right angle is the included angle between hypotenuse and the other side. AAS is ASA in practice once the third angle is fixed. Do not quote SSA as a general congruence test; it is the ambiguous case. In proofs, write the test name and the matching facts: “AB = DE (given), angle A = angle D (alternate angles), AC = DF (given), so SAS.” Each line needs a reason. Congruent triangles give equal corresponding parts you can use in later steps — equal angles unlock parallel line facts, equal sides unlock isosceles facts.
Go deeper
Similar triangles and scale factor
Triangles ABC and DEF are similar with A matching D, B matching E, C matching F. If AB = 4 cm and DE = 10 cm, scale factor small to large is 10 ÷ 4 = 2.5. Then BC = 6 cm implies EF = 6 × 2.5 = 15 cm. If AC = 5 cm and DF is unknown: DF = 5 × 2.5 = 12.5 cm. Always write which triangles are similar and which vertices match before you multiply. Parallel line diagrams: DE parallel to BC in triangle ABC creates triangle ADE similar to triangle ABC with scale factor AD ÷ AB. The shared angle at A and alternate angles give equal angles; sides along the same rays are proportional. That ratio is the scale factor for every unknown on the smaller triangle.
Go deeper
Area and volume with similarity
If lengths scale by k, areas scale by k² because area is two-dimensional. A rectangle 2 cm by 3 cm has area 6 cm². Scale lengths by 3: 6 cm by 9 cm, area 54 cm², and 54 ÷ 6 = 9 = 3². Volume scales by k³. A cube of side 2 cm has volume 8 cm³. Side 6 cm (k = 3) gives volume 216 cm³, and 216 ÷ 8 = 27 = 3³. Exam questions: “similar solids, height 10 cm and 15 cm, smaller volume 40 cm³, find larger volume.” Scale factor 15 ÷ 10 = 1.5, volume factor 1.5³ = 3.375, larger volume = 40 × 3.375 = 135 cm³. For area of similar shapes, use k². Mixing k, k² and k³ is the standard error — label what you are scaling: length, area, or volume.
See the idea in action
Triangles ABC and PQR are similar. AB = 6 cm, BC = 8 cm, PQ = 9 cm. Find QR and the scale factor from ABC to PQR. Step 1: AB corresponds to PQ (given matching). Scale factor = PQ ÷ AB = 9 ÷ 6 = 1.5. Step 2: BC corresponds to QR. QR = 8 × 1.5 = 12 cm. Step 3: Check: matching sides are in ratio 1 : 1.5. 6 : 9 = 8 : 12. If area of ABC is 20 cm², area of PQR = 20 × 1.5² = 20 × 2.25 = 45 cm².
Exam technique
Turn knowledge into marks
Write the vertex correspondence (A ↔ P, B ↔ Q) before you multiply. A scale factor from mis-paired sides poisons every length after.
Common mistakes
Do not give these marks away
- 01
Pairing the shortest side of one triangle with the longest of the other.
- 02
Using scale factor k on area instead of k², or on volume instead of k³.
- 03
Claiming congruence from SSA without checking whether the triangles are genuinely congruent.
Two similar triangles have corresponding sides 4 cm and 10 cm. The larger triangle’s area is how many times the smaller’s?
A2.5
B6.25
C25
D100
Show the answer
6.25. Linear scale factor 10 ÷ 4 = 2.5. Area scales by k², so 2.5² = 6.25. 2.5 is k itself. 25 is 5² from mis-pairing. 100 is k⁴.
Quick questions
If this is the bit you searched
What is the difference between congruent and similar?
Congruent shapes are identical in size and shape. Similar shapes have the same angles but sides in proportion — one is a scaled copy of the other.
How do I find the scale factor?
Divide a length on the larger shape by the corresponding length on the smaller. Write which sides correspond before you divide.
Why is area scale factor k²?
Area uses two lengths. If each length scales by k, area scales by k × k = k². Volume scales by k³.
Which congruence tests work for all triangles?
SSS, SAS, ASA and RHS (for right-angled triangles). SSA is not a general congruence test.