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Similar Triangles — Definition, Formula & Examples

Similar triangles are triangles that have the same shape — all three pairs of corresponding angles are equal and all three pairs of corresponding sides are in the same ratio. They can differ in size but not in shape.

Two triangles △ABC\triangle ABC and △DEF\triangle DEF are similar, written △ABC∼△DEF\triangle ABC \sim \triangle DEF, if and only if their corresponding angles are congruent (∠A=∠D\angle A = \angle D, ∠B=∠E\angle B = \angle E, ∠C=∠F\angle C = \angle F) and their corresponding sides are proportional: ABDE=BCEF=ACDF\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}. This common ratio is called the scale factor.

Key Formula

ABDE=BCEF=ACDF=k\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k
Where:
  • AB,BC,ACAB, BC, AC = Side lengths of the first triangle
  • DE,EF,DFDE, EF, DF = Corresponding side lengths of the second triangle
  • kk = Scale factor — the constant ratio between corresponding sides

How It Works

To show two triangles are similar, you can use one of three tests: AA (two pairs of equal angles), SAS similarity (two pairs of proportional sides with the included angle equal), or SSS similarity (all three pairs of sides in the same ratio). Once you know triangles are similar, you set up a proportion between corresponding sides to find an unknown length. The key step is matching corresponding vertices correctly — the order in which you name the triangles tells you which sides and angles correspond. For example, if △ABC∼△DEF\triangle ABC \sim \triangle DEF, then side ABAB corresponds to side DEDE, not to DFDF.

Worked Example

Problem: Triangle ABC has sides AB = 6, BC = 8, and AC = 10. Triangle DEF is similar to triangle ABC with a scale factor of 2. Find the side lengths of triangle DEF.
Step 1: Identify corresponding sides. Since the triangles are similar with the same vertex order, AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF.
Step 2: Multiply each side of triangle ABC by the scale factor k = 2.
DE=6×2=12,EF=8×2=16,DF=10×2=20DE = 6 \times 2 = 12, \quad EF = 8 \times 2 = 16, \quad DF = 10 \times 2 = 20
Step 3: Verify the ratios are equal.
DEAB=126=2,EFBC=168=2,DFAC=2010=2  ✓\frac{DE}{AB} = \frac{12}{6} = 2, \quad \frac{EF}{BC} = \frac{16}{8} = 2, \quad \frac{DF}{AC} = \frac{20}{10} = 2 \; \checkmark
Answer: The sides of triangle DEF are DE = 12, EF = 16, and DF = 20.

Another Example

Problem: In the figure, triangle PQR has angles P = 50° and Q = 60°. Triangle XYZ has angles X = 50° and Z = 70°. Are the triangles similar? If so, find the missing side YZ given PQ = 9, QR = 12, and XY = 6.
Step 1: Find all angles. In triangle PQR, angle R = 180° − 50° − 60° = 70°. In triangle XYZ, angle Y = 180° − 50° − 70° = 60°.
∠R=70°,∠Y=60°\angle R = 70°, \quad \angle Y = 60°
Step 2: Match corresponding angles: angle P = angle X = 50°, angle Q = angle Y = 60°, angle R = angle Z = 70°. By AA similarity, the triangles are similar.
△PQR∼△XYZ\triangle PQR \sim \triangle XYZ
Step 3: Set up a proportion using corresponding sides PQ ↔ XY and QR ↔ YZ.
PQXY=QRYZ  ⟹  96=12YZ\frac{PQ}{XY} = \frac{QR}{YZ} \implies \frac{9}{6} = \frac{12}{YZ}
Step 4: Solve for YZ by cross-multiplying.
YZ=12×69=8YZ = \frac{12 \times 6}{9} = 8
Answer: Yes, the triangles are similar, and YZ = 8.

Why It Matters

Similar triangles are central to middle-school and high-school geometry courses and appear on nearly every standardized math exam. They are the foundation of trigonometry — sine, cosine, and tangent ratios only work because all right triangles with the same acute angle are similar. Architects, surveyors, and engineers routinely use similar triangles to calculate distances that are difficult to measure directly, such as the height of a building from its shadow.

Common Mistakes

Mistake: Matching sides by length instead of by position. Students pair the longest side of one triangle with the shortest side of another.
Correction: Always match sides by the angles they are opposite or by the vertex order given in the similarity statement. If △ABC∼△DEF\triangle ABC \sim \triangle DEF, then side ABAB (opposite CC) corresponds to side DEDE (opposite FF).
Mistake: Setting up the proportion with sides from both triangles on the same side of the equation, such as writing AB/BC = DE/EF incorrectly as AB/DE = EF/BC.
Correction: Keep corresponding sides in matching positions. A correct proportion pairs one triangle's sides in the numerators and the other's in the denominators: ABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF}.

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