What is the difference between similar and congruent triangles?

Congruent triangles are identical in both shape and size — a scale factor of exactly 1. Similar triangles share the shape and all their angles but may be any size. Every congruent pair is also similar; the reverse is not true.

How many angles do I need to prove similarity?

Two. If two pairs of angles match, the third must match as well since angles in a triangle sum to 180°. That is the AA test, and it is usually the easiest to satisfy because it needs no side lengths.

How do I know which sides correspond?

Corresponding sides sit opposite equal angles. If the problem states \triangle ABC \sim \triangle DEF, the vertex order does the work for you: AB pairs with DE, BC with EF, AC with DF.

What is the scale factor?

The number every side of one triangle is multiplied by to produce the other. A factor above 1 means an enlargement, below 1 a reduction. It is found by dividing any pair of corresponding sides.

Why do areas scale by k^2?

Because area is a product of two lengths, and both get multiplied by k. So the area is multiplied by k \cdot k = k^2. Volumes, being three lengths, scale by k^3.

How is this connected to trigonometry?

Directly. All right triangles with the same acute angle are similar, so the ratio of any two sides depends only on that angle — which is exactly what \sin, \cos and \tan record. Similarity is what makes trig ratios well-defined.

You can start solving similar triangle problems for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

Similar Triangles Calculator

Enter two corresponding sides to get the scale factor, then any missing length — with the proportion written out. Free.

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On this page01Summary 02Definition 03Formulas 04Methods 05Examples 06Practice 07Mistakes 08FAQ

Summary

Two triangles are similar when their corresponding angles are equal, and their corresponding sides are then in one constant ratio throughout.

Establish similarity with AA, SAS or SSS, write the corresponding sides in the same order on both sides of a proportion, and cross-multiply to find the missing length.

Similar is not congruent: equal angles fix the shape but say nothing about the size, so you cannot conclude that corresponding sides are equal — only that they share a scale factor.

01Definition

What are similar triangles?

Two triangles are similar when they have the same shape but not necessarily the same size — one is a scaled copy of the other. All three pairs of angles are equal, and all three pairs of corresponding sides are in the same ratio. The symbol is $\sim$ , as in $\triangle ABC \sim \triangle DEF$ .

That constant ratio is the scale factor. If one triangle’s sides are $3, 4, 5$ and the other’s are $9, 12, 15$ , the scale factor is $3$ — every side has been tripled, and every angle is untouched. Scaling changes size, never shape.

Similarity is the engine behind a surprising amount of math. It is why trigonometric ratios work at all — every right triangle with a $30°$ angle is similar to every other, so $\sin 30°$ is a single fixed number regardless of how big the triangle is.

Where you'll actually use this

Measuring a tree or building by comparing its shadow with a metre stick’s shadow.
Map and blueprint scales: 1:500 is a scale factor, and areas on the plan scale by 1:250 000.
Photography and projection: image size grows in proportion to distance from the lens.
Model kits and 3D printing, where a scale change multiplies material use by $k^3$ .
Indirect measurement in surveying, where an inaccessible distance is found from an accessible similar triangle.

Similar: Same shape, possibly different size — equal angles and proportional sides.
Scale factor: The constant ratio between corresponding sides of two similar figures.
Corresponding sides: A pair of sides, one from each triangle, sitting opposite equal angles.
AA similarity: Two pairs of equal angles are sufficient to prove similarity.
Geometric mean: For $\frac{a}{h} = \frac{h}{b}$ , the value $h = \sqrt{ab}$ — appears in the altitude-to-hypotenuse case.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Similarity statement

\triangle ABC \sim \triangle DEF

Vertex order matters: A matches D, B matches E, C matches F.

Side proportion

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

k is the scale factor — the same for all three pairs.

AA test

\angle A = \angle D,\; \angle B = \angle E

Two equal angles are enough; the third follows automatically.

SSS test

\frac{a}{d} = \frac{b}{e} = \frac{c}{f}

All three side ratios equal — no angle information needed.

SAS test

\frac{a}{d} = \frac{b}{e},\; \angle \text{between them equal}

The equal angle must be the one enclosed by the two proportional sides.

Area ratio

\frac{A_1}{A_2} = k^2

Areas scale by the square of the scale factor, not by k itself.

03Methods

How to solve similar triangles

Pick the method that fits the problem in front of you.

When to use it: One pair of corresponding sides is fully known — the quickest route to any other missing side.

1Divide a known side of the large triangle by its partner in the small one: $\frac{9}{3} = 3$ .
2Multiply any other small-triangle side by that factor to get its partner: $5 \cdot 3 = 15$ .
3Going the other way, divide instead: a large side of $12$ maps to $12 \div 3 = 4$ .
4Check the factor is consistent across a second pair of sides before trusting it.

04Worked examples

Similar Triangles, solved step by step

From easy to hard — pick a problem to see its full solution.

\triangle ABC \sim \triangle DEF,\; AB = 3,\; DE = 9,\; BC = 5,\; EF = ?

1Scale factor from the known pair: $\frac{9}{3} = 3$ .
2Apply it to $BC$ : $EF = 5 \cdot 3 = 15$ .
3Check: $\frac{3}{9} = \frac{5}{15}$ ✓.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\frac{5}{x} = \frac{15}{21}

Problem 2

\triangle ABC \sim \triangle PQR,\; AB = 8,\; PQ = 12,\; AC = 6,\; PR = ?

Problem 3

\text{Are } 3, 5, 7 \text{ and } 9, 15, 20 \text{ similar?}

Problem 4

\text{Similar triangles, } k = 4. \text{ Small perimeter } 15. \text{ Large perimeter?}

Problem 5

\text{Two similar triangles have areas } 16 \text{ and } 49. \text{ Find } k.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Pairing sides by position on the page rather than by corresponding vertices.	One triangle is often drawn rotated or reflected, so the "left side" of each is not the matching pair.	Use the similarity statement’s vertex order — in $\triangle ABC \sim \triangle DEF$ , $AB$ pairs with $DE$ . If no statement is given, match sides by the equal angles they sit opposite.
Scaling area by $k$ instead of $k^2$ .	The side rule is learned first and gets applied to every quantity in the problem.	Lengths scale by $k$ , areas by $k^2$ , volumes by $k^3$ — match the power to the number of dimensions. Doubling every side quadruples the area.
Flipping one fraction so the proportion mixes small over large with large over small.	The proportion gets written left-to-right off the diagram without tracking which triangle each length belongs to.	Fix a convention — small on top on both sides — and label each length with its triangle before writing the equation. A missing side that comes out on the wrong side of 1 is the tell.
Confusing similar with congruent.	Both involve matching triangles, and the tests share the letters AA, SSS, SAS.	Congruent means identical — scale factor exactly $1$ . Similar allows any scale factor. SSS congruence needs equal sides; SSS similarity only needs proportional ones.
In nested triangles, using a part of the large side as the whole side.	The diagram labels the two segments the parallel line creates, and the nearer one gets used directly.	The large triangle’s side is the full length from the apex — usually the sum of both labelled segments. Add them before writing the proportion.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ