Similar Triangles Explained: Definition, Rules, Formulas & Examples

In geometry, “similar” means two figures share the same shape but not always the same size. Similar triangles have equal angles and proportional sides—one is a scaled version of the other. Common mistakes include confusing similar with congruent, misunderstanding side proportions, and mixing up the area ratio. Mastering these ideas is a key part of Pre-Algebra math and lays the groundwork for geometry proofs and real-world problem solving.

In this guide, we’ll explain the definition, similarity rules, formulas, and step-by-step examples.

What Does “Similar” Mean?

In geometry, the word similar means exactly the same in shape, but might be different in size. Two figures are similar if one can be transformed into the other by flipping, turning, sliding, with resizing.

Similar vs. Congruent

Congruent shapes are a special kind of similar shapes. When two shapes are congruent, they have exactly the same in size and shape. That means we don’t need to resize, just slide, flip, or turn.

What Are Similar Triangles?

Similar triangles have the same angles and their matching sides are in the same ratio. One triangle is a resized version of the other.

This idea is often written as: △ABC ∼ △DEF

How Can You Tell If Two Triangles Are Similar?

You do not always need to compare every angle and every side. Three common similarity rules can prove that two triangles are similar.

AA Similarity

AA stands for Angle-Angle.

If two angles in one triangle are equal to two corresponding angles in another triangle, the triangles are similar.

The third pair of angles must also be equal because the three angles in every triangle add up to 180°.

For example:

• ∠A = ∠D
• ∠B = ∠E

Therefore: △ABC ∼ △DEF by AA similarity.

AA is one of the most commonly used reasons for proving that two triangles are similar.

SAS Similarity

SAS stands for Side-Angle-Side.

Two triangles are similar by SAS when:

• Two pairs of corresponding sides are proportional.
• The included angles between those sides are equal.

For example, suppose: AB ÷ DE = AC ÷ DF

and: ∠A = ∠D

Because ∠A and ∠D are the angles between the two pairs of proportional sides, the triangles are similar by SAS.

SSS Similarity

SSS stands for Side-Side-Side.

Two triangles are similar by SSS when all three pairs of corresponding sides have the same ratio.

For example: AB ÷ DE = BC ÷ EF = AC ÷ DF

Therefore: △ABC ∼ △DEF by SSS similarity.

Unlike congruent triangles, the corresponding sides do not need to have equal lengths. They only need to be proportional.

What Is the Similar Triangles Formula?

The main formula for similar triangles compares their corresponding sides:

AB : DE = BC : EF = AC : DF

You can use this formula to find a missing side length.

For example, suppose △ABC ∼ △DEF and:

• AB = 4
• DE = 6
• BC = 10
• EF is unknown

Write a proportion using corresponding sides:

4 : 6 = 10 : EF

Cross-multiply:

4 × EF = 6 × 10

4 × EF = 60

EF = 15

Therefore, the missing side length is 15.

What Is the Scale Factor of Similar Triangles?

The scale factor tells you how much one triangle has been enlarged or reduced.

Scale factor = new side length ÷ corresponding original side length

For example, if a side increases from 4 units to 10 units:

Scale factor = 10 ÷ 4 = 2.5

Every side in the larger triangle is 2.5 times the length of its corresponding side in the smaller triangle.

If the scale factor is:

• Greater than 1, the triangle is enlarged.
• Between 0 and 1, the triangle is reduced.
• Equal to 1, the triangles are congruent.

Be consistent about which triangle is the original and which is the new triangle. Reversing the order will give the reciprocal scale factor.

Side, Perimeter, and Area Ratios

When two triangles are similar, their corresponding side lengths and perimeters have the same ratio.

If the side ratio is 2:3, then the perimeter ratio is also 2:3.

However, the area ratio is the square of the side ratio.

Area ratio = (side ratio)²

If the side ratio is 2:3:

Area ratio = 2²:3² = 4:9

This means that if the smaller triangle has an area of 20 square units, the larger triangle has an area of:

20 × 9 ÷ 4 = 45 square units

How to Solve Similar Triangles Step by Step

Use the following process when solving for a missing side in two similar triangles.

Step 1: Match the Corresponding Vertices

Use the similarity statement to determine which vertices and sides correspond.

If: △ABC ∼ △DEF

then:

• A corresponds to D.
• B corresponds to E.
• C corresponds to F.

Step 2: Match the Corresponding Sides

From the same statement:

• AB corresponds to DE.
• BC corresponds to EF.
• AC corresponds to DF.

Do not compare sides that are not corresponding.

Step 3: Write a Proportion

Place corresponding sides in the same positions:

AB : DE = BC : EF

You may use a different pair of sides, but the order must remain consistent.

Step 4: Solve for the Missing Length

Cross-multiply and solve the resulting equation.

Step 5: Check the Scale Factor

Confirm that all corresponding side lengths follow the same scale factor.

If the larger triangle has some sides that appear shorter than the corresponding sides of the smaller triangle, check whether the sides were matched correctly.

Similar Triangles Examples with Solutions

Example 1: Finding a Missing Angle

Problem: Given that △ABC ∼ △DEF, ∠A = 45°, and ∠B = 90°, what is the measure of ∠F?

Solution:

Because △ABC ∼ △DEF, all corresponding angles are equal.

The order of the similarity statement tells us that ∠C corresponds to ∠F.

First, find ∠C using the triangle angle sum:

∠C = 180° − (∠A + ∠B)

∠C = 180° − (45° + 90°)

∠C = 45°

Because ∠C = ∠F: ∠F = 45°

Answer: The measure of ∠F is 45°.

Example 2: Finding the Area of a Similar Triangle

Problem: Given that △XYZ ~ △MNO, and the ratio of their sides is 2:3. The area of △XYZ is 20 cm². What is the area of △MNO?

Solution:

The area ratio is the square of the side ratio:

Area ratio = 2²:3² = 4:9

This means the area of △XYZ is 4/9 of the area of △MNO.

Let A represent the area of △MNO: 20 ÷ A = 4 ÷ 9

Cross-multiply:

4A = 20 × 9

4A = 180

A = 45

Answer: The area of △MNO is 45 cm².

Example 3: Finding a Missing Side Length

Problem: Given that △ABC ∼ △DEF: AB = 6, DE = 9, BC = 8. Find EF.

Solution:

AB corresponds to DE, and BC corresponds to EF.

Write a proportion: 6 : 9 = 8 : EF

Cross-multiply:

6 × EF = 9 × 8

6 × EF = 72

EF = 12

Answer: EF = 12.

You can check the result using the scale factor: Scale factor = 9 ÷ 6 = 1.5

Then: 8 × 1.5 = 12

The result is correct.

Example 4: Identifying the Similarity Rule

Problem: Two triangles have the following angle measures:

• Triangle 1: 40°, 60°, and 80°
• Triangle 2: 40°, 60°, and 80°

Are the triangles similar?

Solution:

The triangles have two pairs of equal corresponding angles:

• 40° = 40°
• 60° = 60°

Therefore, the triangles are similar by AA similarity.

Answer: Yes, the triangles are similar by AA.

5 Common Mistakes with Similar Triangles

Matching the Wrong Sides

Always follow the order of the similarity statement.

If △ABC ∼ △DEF, then AB corresponds to DE—not EF or DF.

Using the Side Ratio as the Area Ratio

If the side ratio is 2:3, the area ratio is not 2:3.

The correct area ratio is:

2²:3² = 4:9

Assuming Similar Triangles Are Congruent

Similar triangles have the same shape, but they may have different sizes. Congruent triangles must have both the same shape and the same size.

Comparing Sides in Different Orders

The following proportions are consistent:

AB ÷ DE = BC ÷ EF

But this proportion is inconsistent:

AB ÷ DE = EF ÷ BC

When one ratio compares the first triangle to the second, every other ratio must follow the same order.

Relying Only on Appearance

Triangles may appear similar in a diagram without actually being similar. Use AA, SAS, or SSS to justify the similarity.

Summary: Key Takeaways About Similar Triangles

• Similar triangles have the same shape but may have different sizes.
• Their corresponding angles are equal.
• Their corresponding sides are proportional.
• The similarity statement shows which vertices and sides correspond.
• AA, SAS, and SSS can be used to prove that two triangles are similar.
• The scale factor compares corresponding side lengths.
• The perimeter ratio equals the side ratio.
• The area ratio equals the square of the side ratio.
• Similar triangles can be used to find missing side lengths, angles, perimeters, and areas.
• Similar vs. Congruent
  • Congruent: same shape and same size
  • Similar: same shape, size may be different

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• Congruent Triangles Explained: Geometry Rules & Examples

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