Two triangles are similar (same shape, possibly different size) if they satisfy one of 3 tests: AA (two pairs of equal angles), SSS (all three pairs of sides in the same ratio), or SAS (two pairs of sides in the same ratio with the included angle equal). The constant ratio between corresponding sides is the scale factor, k. Once you know k: lengths scale by k, areas scale by k², and volumes scale by k³ — the rule examiners trap students on most.
Similar triangles have the same shape: corresponding angles are equal and corresponding sides are all in the same ratio, but the two triangles can be different sizes. Congruent triangles are the stricter case — same shape and same size, i.e. similar with a scale factor of exactly 1. If a question gives you sides of different lengths that are still in proportion, you are looking at similarity, not congruence. (If you need the size-must-match case, see the companion guide on congruence tests, SSS, SAS, AAS, RHS.)
AA — Angle, Angle
Two angles of one triangle equal two angles of the other. Because angles in a triangle sum to 180°, the third pair is then automatically equal too.
SSS — Side, Side, Side (in proportion)
All three pairs of corresponding sides are in the same ratio. Here every side of the larger triangle is 1.5× the smaller, so k = 1.5.
SAS — Side, Angle, Side (in proportion)
Two pairs of sides are in the same ratio and the angle between them (the included angle) is equal in both triangles.
✗ Watch the difference from congruence. For similarity, AA is enough — you do not need a side. For congruence, AA (or AAA) is never enough, because equal angles fix the shape but not the size. Same three letters, opposite verdict: AA proves similar, but not congruent.
Most similarity questions are really "find the missing side" in disguise. The method is always the same: prove similar, find k from a pair of known corresponding sides, then multiply or divide.
Worked example
k = QR / BC = 9 / 6 = 1.5.x = PQ = k × AB = 1.5 × 5 = 7.5.x = 7.5. Always match corresponding sides (the side opposite the equal angle), not just "the sloping one".Set it up as a ratio, every time: write PQ/AB = QR/BC before substituting numbers. Putting corresponding sides in matching positions (big-over-small on both sides, or small-over-big on both) stops the most common error — accidentally flipping one fraction and dividing when you should multiply.
This is the single most tested idea in the whole topic, and the one that quietly loses the most marks. It is the same idea behind a map scale, where a length ratio of 1 : n becomes an area ratio of 1 : n². When two figures are similar with scale factor k, the three quantities scale differently:
Length
× k
Sides, perimeters, heights, radii — anything one-dimensional.
Area
× k²
Surface area and any area. Double the lengths → 4× the area.
Volume
× k³
Volume, capacity, and mass (same material). Double the lengths → 8× the volume.
Worked example — area and volume scale factor
k = 15 / 10 = 1.5.k³: 400 × 1.5³ = 400 × 3.375 = 1350 ml.k² = 1.5² = 2.25.Common mistake
Multiplying the volume by k instead of k³ (here, giving 400 × 1.5 = 600 ml). If the question is about capacity, volume, or mass, it is always k³; if it is about area or surface area, it is always k². Only actual lengths use k on its own.
The classic AA setup: a line drawn parallel to one side of a triangle cuts the other two sides and creates a smaller triangle similar to the whole. The shared apex angle plus a pair of corresponding angles (from the parallel lines) gives you AA instantly — no side lengths needed to prove similarity.
What is the difference between similar and congruent triangles?
Similar means same shape, so corresponding angles are equal and corresponding sides are in the same ratio — but the sizes can differ. Congruent means same shape and same size, which is just the special case where the scale factor is 1. Every congruent pair is also similar, but not the other way round.
Why is AA enough for similarity but not for congruence?
Two equal angles fix the shape of a triangle completely (the third angle follows automatically), which is all similarity needs. But equal angles say nothing about size — two triangles with identical angles can be any scale of each other — so AA can never prove congruence, which also requires the sizes to match.
If lengths scale by k, why does area scale by k² and volume by k³?
Area is made from two length dimensions multiplied together, so it grows by k × k = k². Volume is made from three, so it grows by k × k × k = k³. That is why doubling the lengths of a shape multiplies its area by 4 and its volume by 8, not by 2.
How do I know which sides are "corresponding"?
Corresponding sides sit opposite equal angles, and they appear in the same position when you name the triangles in matching order. If triangle ABC is similar to triangle PQR, then AB corresponds to PQ, BC to QR, and CA to RP. Naming the vertices in the right order first makes the ratios line up automatically.
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Mr. Gan turns "find the missing side" into a repeatable 3-step routine, so similarity questions stop costing easy marks.
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