Two triangles are similar when they have the same shape but not necessarily the same size. One is a scaled copy of the other. Their angles are equal and their sides are in the same ratio.
What similar means
If △ABC ~ △DEF (the symbol ~ means “is similar to”), then:
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
- AB/DE = BC/EF = AC/DF
The common ratio is the scale factor. The order of the letters matters: A matches D, B matches E, C matches F. Read the corresponding sides straight off the names.
Three ways to prove triangles are similar
| Test | What you need | Example |
|---|---|---|
| AA | Two pairs of equal angles | Both have a 90° angle and a 35° angle |
| SAS | Two pairs of sides in the same ratio, and the angle between them equal | 4/6 = 6/9, with a 50° angle between each pair |
| SSS | All three pairs of sides in the same ratio | 3, 4, 5 and 6, 8, 10 |
AA is the one you’ll use most. Two angles is enough because the third angle is forced: every triangle’s angles add to 180°.
SSA is not a similarity test
Two proportional sides and an angle that is not between them do not prove similarity. The angle has to be the one the two sides form.
Finding a missing side
Once you know the triangles are similar, set up a proportion with corresponding sides and cross-multiply. The ratios and proportions guide covers the algebra.
Example △PQR ~ △XYZ with PQ = 6, QR = 9, XY = 10. Find YZ.
- Match the sides: PQ ↔ XY, QR ↔ YZ.
- Set up: 6/10 = 9/YZ.
- Cross-multiply: 6 · YZ = 90.
- YZ = 15.
Keep the order the same on both sides of the proportion. Small/big = small/big works. Small/big = big/small doesn’t.
Triangles inside triangles
A very common setup: a line parallel to one side cuts across a triangle. The small triangle at the top is similar to the whole triangle by AA, because the parallel line makes equal corresponding angles and the top angle is shared.
Example In △ABC, DE ∥ BC, with D on AB and E on AC. AD = 4, DB = 6, DE = 5. Find BC.
- △ADE ~ △ABC.
- AB = AD + DB = 10. Compare AD/AB, not AD/DB.
- 4/10 = 5/BC.
- BC = 12.5.
Using the piece instead of the whole side
The side of the big triangle is AB = 10, not DB = 6. Draw the two triangles separately if the overlap is confusing.
Shadows and indirect measurement
Sun rays hit the ground at the same angle for two objects at the same time, and both stand at right angles to the ground. That makes AA similarity, so shadows give you heights you can’t measure directly.
Example A 1.8 m person casts a 2.4 m shadow. At the same time a tree casts a 20 m shadow. How tall is the tree?
- height/shadow is the same for both: 1.8/2.4 = h/20.
- h = 20 × 1.8/2.4 = 15.
- The tree is 15 m tall.
How perimeter, area and volume scale
If the scale factor is k:
- Perimeters (and any other lengths) scale by k.
- Areas scale by k².
- Volumes of similar solids scale by k³.
Example Two similar triangles have sides in the ratio 3:5. The smaller one has area 18 cm². Find the larger area.
- The area ratio is 3²:5² = 9:25.
- 18/A = 9/25, so A = 50 cm².
Similar right triangles
Drawing the altitude to the hypotenuse of a right triangle splits it into two smaller triangles, and all three are similar. This is one proof of the Pythagorean theorem, and it is also why trig ratios work: every right triangle with a 30° angle is similar, so sin 30° is always the same number.
Your turn
A triangle has angles 40° and 75°. Another has angles 75° and 65°. Are they similar?
Show answer
Yes. The first triangle’s third angle is 65°, so both have 40°, 75° and 65° (AA).
Are triangles with sides 5, 7, 9 and 10, 14, 18 similar?
Show answer
Yes, by SSS. Every ratio is 2.
△ABC ~ △DEF, AB = 8, DE = 12, AC = 10. Find DF.
Show answer
15. 8/12 = 10/DF.
In △ABC, DE ∥ BC, AD = 3, DB = 9, BC = 16. Find DE.
Show answer
4. AD/AB = 3/12 = DE/16.
A 2 m pole casts a 3 m shadow while a building casts a 45 m shadow. How tall is the building?
Show answer
30 m.
Two similar triangles have perimeters 12 and 30. The smaller has area 8. Find the larger area.
Show answer
50. The scale factor is 2.5, so the area factor is 6.25.