Maths
What are similar triangles, and how do you find a missing side?
Quick answer
Similar triangles have the same shape but not always the same size. Their angles are equal and their sides are in the same ratio, so one is an enlargement of the other.
In one sentence
Two triangles are similar when one is an exact enlargement (or reduction) of the other: same angles, sides in the same ratio.
Explained
Take a photo of a triangle on your phone and zoom in. The triangle gets bigger, but its shape does not change: the angles stay the same and every side grows by the same amount. The zoomed triangle and the original are similar. Similar triangles are everywhere: a tree and its shadow, a ramp and its supports, a map and the real land.
tree
sun
ruler
Two triangles are similar when:
- their corresponding angles are equal, and
- their corresponding sides are in the same ratio.
That ratio is the scale factor, usually called k. If triangle ABC is similar to triangle DEF, we write △ABC ~ △DEF. The order of the letters matters: A matches D, B matches E and C matches F. Then
DE ÷ AB = EF ÷ BC = DF ÷ AC = k
This is the "formula" for similar triangles: one fraction for each pair of matching sides, all equal. Working with it is really working with ratio.
You do not have to check everything to prove two triangles are similar. Any one of these tests is enough:
- AA (angle–angle): two pairs of angles are equal. The third pair must then be equal too, because the angles of a triangle add up to 180°.
- SSS (side–side–side): all three pairs of sides are in the same ratio.
- SAS (side–angle–side): two pairs of sides are in the same ratio, and the angles between them are equal.
A very common case is a triangle with a line drawn parallel to one side. The small triangle at the top and the whole triangle share one angle, and the parallel line makes the other angles equal (corresponding angles). So they are similar by AA. This is the idea behind the intercept theorem, which is called Thales' theorem in France, Spain, Italy and other countries.
Lengths grow by k, but areas grow by k². Double every side and the area becomes four times bigger, because both the base and the height doubled.
Examples
| Triangle 1 | Triangle 2 | Test | Similar? |
|---|---|---|---|
| sides 3, 4, 5 | sides 6, 8, 10 | 6/3 = 8/4 = 10/5 = 2 | yes, k = 2 |
| angles 40°, 60° | angles 60°, 80° | third angles: 80° and 40° | yes (AA) |
| sides 4, 6, 8 | sides 6, 9, 13 | 6/4 = 1.5, 9/6 = 1.5, 13/8 = 1.625 | no |
| sides 5 and 8, angle 50° between | sides 10 and 16, angle 50° between | 10/5 = 16/8 = 2 | yes (SAS) |
Finding a missing side. △ABC ~ △DEF with AB = 4 cm, BC = 6 cm and DE = 10 cm. The scale factor is k = 10 ÷ 4 = 2.5, so EF = 6 × 2.5 = 15 cm.
Shadows. A 1.5 m tall person casts a 2 m shadow. At the same moment a tree casts a 12 m shadow. The sun's rays hit both at the same angle, so the triangles are similar: height ÷ shadow is the same. Tree height = 1.5 × 12 ÷ 2 = 9 m.
Common mistakes
- Matching the wrong sides. Fix: pair sides by the angles they are opposite: shortest with shortest, longest with longest. Redraw the triangles the same way round if they are turned or flipped.
- Adding instead of multiplying. Going from 4 to 10 is "× 2.5", not "+ 6". Every side is multiplied by the same scale factor.
- Using the scale factor for area. If k = 3, the area is 3² = 9 times bigger, not 3.
- Using part of a side in a nested triangle. With a parallel line inside a triangle, compare the small triangle with the whole big triangle: use AB, not just the piece DB.
Step by step
Step by step
- Check they are similarUse AA, SSS or SAS. Often two equal angles or a parallel line is enough.
- Match the cornersDecide which vertex matches which. Redraw the triangles side by side, the same way round, if it helps.
- Find the scale factorPick a pair of matching sides you know both of: k = new length ÷ original length.
- Find the missing sideMultiply by k to go from small to big; divide by k to go from big to small.
- CheckThe bigger triangle must have the longer sides, and every ratio should give the same k.
You can also set up a proportion and cross-multiply: if 4/10 = 6/x, then 4x = 60 and x = 15. It is the same calculation as using the scale factor.
Practice questions
Similar triangles
Try each one, then check your answer.
- A triangle with sides 2, 3 and 4 cm is enlarged by a scale factor of 3. What are the new sides?
Show answer
2 × 3, 3 × 3, 4 × 3 = 6, 9 and 12 cm - Are triangles with sides 5, 12, 13 and 10, 24, 26 similar?
Show answer
10/5 = 24/12 = 26/13 = 2: yes, with k = 2 - Are triangles with sides 3, 5, 6 and 6, 10, 11 similar?
Show answer
6/3 = 2 and 10/5 = 2, but 11/6 ≈ 1.83: no - One triangle has angles 50° and 70°. Another has angles 70° and 60°. Are they similar?
Show answer
The first triangle's third angle is 180° − 50° − 70° = 60°. Both have 50°, 60°, 70°: yes (AA) - One triangle has angles 45° and 85°. Another has angles 45° and 55°. Are they similar?
Show answer
Third angles: 50° and 80°. The angles are 45°, 50°, 85° and 45°, 55°, 80°: no - △ABC ~ △PQR, with AB = 6 cm and PQ = 9 cm. What is the scale factor from ABC to PQR?
Show answer
9 ÷ 6 = 1.5 - In the same triangles, BC = 8 cm. Find QR.
Show answer
8 × 1.5 = 12 cm - In the same triangles, PR = 15 cm. Find AC.
Show answer
15 ÷ 1.5 = 10 cm - Two similar triangles have matching sides of 4 cm and 10 cm. The smaller one has a perimeter of 15 cm. Find the perimeter of the larger one.
Show answer
k = 10 ÷ 4 = 2.5, so 15 × 2.5 = 37.5 cm - Two similar triangles have a scale factor of 3. The smaller one has an area of 5 cm². Find the area of the larger one.
Show answer
5 × 3² = 45 cm² - Two similar triangles have areas of 16 cm² and 36 cm². What is the ratio of their matching sides?
Show answer
√16 : √36 = 4 : 6 = 2 : 3 - Triangles with sides 6 cm and 9 cm and an angle of 40° between them, and with sides 4 cm and 6 cm and an angle of 40° between them. Similar?
Show answer
6/4 = 9/6 = 1.5 and the included angles are equal: yes (SAS) - In triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 4 cm, AB = 10 cm and DE = 6 cm. Find BC.
Show answer
△ADE ~ △ABC (AA), k = 10 ÷ 4 = 2.5, so BC = 6 × 2.5 = 15 cm - In the same figure, AE = 5 cm. Find AC and EC.
Show answer
AC = 5 × 2.5 = 12.5 cm, so EC = 12.5 − 5 = 7.5 cm - In a right-angled triangle, the altitude from the right angle splits the hypotenuse into pieces of 4 cm and 9 cm. Find the altitude h.
Show answer
The two smaller triangles are similar, so h/4 = 9/h. Then h² = 36: h = 6 cm
Word problems
Similar triangles in real life
Try each one, then check your answer.
- A 1.6 m tall student casts a 2 m shadow. At the same time a flagpole casts a 15 m shadow. How tall is the flagpole?
Show answer
1.6 ÷ 2 = h ÷ 15, so h = 1.6 × 15 ÷ 2 = 12 m - A ramp rises steadily. A support post 0.5 m tall stands 2 m (measured along the ground) from the bottom of the ramp. The ramp ends 8 m from the bottom. How high is the top?
Show answer
0.5 ÷ 2 = h ÷ 8, so h = 2 m - You put a mirror on the ground 10 m from a building, then step back until you see the top of the building in it. You are 2 m from the mirror and your eyes are 1.5 m above the ground. How tall is the building?
Show answer
The angles at the mirror are equal, so the triangles are similar: h = 1.5 × 10 ÷ 2 = 7.5 m - A triangular sail is 6 m tall with a 4 m base. A model boat has a similar sail 30 cm tall. How wide is the model sail's base?
Show answer
k = 30 ÷ 600 = 0.05, so 400 cm × 0.05 = 20 cm - A plan shows a triangular garden with sides 3 cm, 4 cm and 5 cm. The real shortest side is 12 m. Find the real longest side and the real area.
Show answer
k = 1200 cm ÷ 3 cm = 400, so the longest side is 5 × 400 cm = 20 m. It is a right-angled triangle with legs 12 m and 16 m: area = ½ × 12 × 16 = 96 m²
Questions people ask
What does "similar" mean in maths?
Similar means the same shape, possibly a different size. Similar triangles have equal angles and sides in the same ratio, like a photo and its enlargement.
What must similar triangles have the same?
The same angles. Their sides do not have to be the same length, but they must all be in the same ratio (the scale factor).
What are the rules for similar triangles?
There are three tests: AA (two equal angles), SSS (all three pairs of sides in the same ratio) and SAS (two pairs of sides in the same ratio with equal angles between them). Any one of them proves similarity.
What is the symbol for similar triangles?
The symbol is ~, so △ABC ~ △DEF means "triangle ABC is similar to triangle DEF". Congruent uses ≅.
What is the difference between similar and congruent triangles?
Similar triangles have the same shape; congruent triangles have the same shape and the same size. Congruent triangles are similar triangles with a scale factor of 1.
Quick quiz
Similar triangles
Check what you know with 4 quick questions.
Question 1 of 4
Two triangles both have angles of 30° and 70°. What can you say?
Practise it
Try it
Practise similar triangles with unlimited questions
Geometry worksheets
Practise it step by step in a full course: Grade 10 Maths
For grown-ups
Related topics
Ratio
A ratio compares two or more amounts, written with a colon. A ratio of 2:3 means for every 2 of one thing there are 3 of the other.
Types of triangles
Triangles are named by their sides (equilateral, isosceles, scalene) or by their angles (acute, right-angled, obtuse). The three angles of any triangle add up to 180 degrees.
Congruent triangles
Congruent triangles are exactly the same shape and size, with all matching sides and angles equal. You can prove it with SSS, SAS, ASA, AAS or RHS.
Transformations: reflection, rotation and translation
Transformations move a shape without changing its size or form. A translation slides it, a reflection flips it over a mirror line, and a rotation turns it around a point.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.