Introduction to Similarity

Learn what it means for two shapes to be similar and how to identify similar figures using proportional sides and equal angles.

Intermediate25 minLesson
Printable worksheet

Definition

Two shapes are similar if they have the same shape but not necessarily the same size. Similar figures have:
  1. 1.Equal corresponding angles - All matching angles are exactly the same
  2. 2.Proportional corresponding sides - All matching sides have the same ratio
We use the symbol ∼\sim to show similarity. If triangle ABCABC is similar to triangle DEFDEF, we write:
△ABC∼△DEF\triangle ABC \sim \triangle DEF
The scale factor is the ratio between corresponding sides of similar figures. If every side of figure B is twice as long as the matching side of figure A, the scale factor is 22.

Try it now

Are all squares similar to each other?

Worked Examples

Triangle ABCABC has angles 50°50°, 60°60°, and 70°70°. Triangle DEFDEF has angles 50°50°, 60°60°, and 70°70°. Are they similar?
1

Compare corresponding angles

Angle A = 50°50° = Angle D Angle B = 60°60° = Angle E Angle C = 70°70° = Angle F → All angles match
2

Apply the AA (Angle-Angle) criterion

If two angles of one triangle equal two angles of another triangle, they are similar → AA criterion satisfied
3

State the conclusion

Since all corresponding angles are equal, the triangles are similar → △ABC∼△DEF\triangle ABC \sim \triangle DEF

Common Mistakes

Confusing similar with congruent

Why it's wrong: Congruent figures are exactly the same size AND shape. Similar figures only need to be the same shape - they can be different sizes.

Correct: All congruent figures are similar (with scale factor 1), but similar figures are not necessarily congruent.

Matching sides incorrectly when comparing figures

Why it's wrong: You must match sides that are in the same position relative to the angles, not just sides of the same length.

Correct: Always identify corresponding angles first, then match the sides that are opposite to those angles.

Using the wrong scale factor direction

Why it's wrong: The scale factor depends on which figure you're scaling from. Going from small to large gives a factor > 1, large to small gives a factor < 1.

Correct: Always clarify: 'Scale factor from figure A to figure B.' If AA is smaller, the scale factor will be greater than 1.

Interactive Visual

Similar triangles

81210PQR121815STU△PQR ∼ △STU

SSSAll three pairs of sides are in the same ratio k.

  • ST : PQ = 12 : 8 = 1.5
  • TU : QR = 18 : 12 = 1.5
  • US : RP = 15 : 10 = 1.5

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Transformation Grid

T(3, 2)
-10-10-8-8-6-6-4-4-2-2224466881010xy(2, 1)(4, 1)(3, 4)(5, 3)(7, 3)(6, 6)
OriginalTransformed

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Two figures are similar if they have:

Why It Matters

Similarity is one of the most useful concepts in geometry and appears everywhere:
  • Architecture: Architects create scale models of buildings before construction
  • Maps: A map is a similar figure to the actual land it represents
  • Photography: Enlarging or reducing a photo creates a similar image
  • Art: Artists use similarity to create depth and perspective
  • Engineering: Engineers test smaller similar models before building full-size structures
Understanding similarity helps you work with proportions, predict measurements, and solve real-world problems involving scale!

Real World Applications

Scale Models in Architecture

Architects build scale models of buildings to show clients and test designs before construction.

Example:

A 1:100 scale model means every 1 cm on the model represents 100 cm (1 meter) on the actual building. A 50-meter-tall building would be 50 cm in the model.

1Try It Yourself

An architect makes a 1:50 scale model of a house. The model is 30 cm long.

How long is the actual house?

Step 1: Write the mathematical expression

Calculate: 30×5030 \times 50

Map Reading

Maps are similar figures to the actual terrain they represent, using a scale to convert distances.

Example:

On a map with scale 1:25000, 4 cm on the map represents 4 × 25000 = 100000 cm = 1 km of actual distance.

2Try It Yourself

On a map, two cities are 8 cm apart. The map scale is 1:500000.

What is the actual distance between the cities in kilometers?

Step 1: Write the mathematical expression

Calculate: 8×5000008 \times 500000

Key Takeaways

  • 1Similar figures have the same shape but not necessarily the same size
  • 2For figures to be similar: corresponding angles must be equal AND corresponding sides must be proportional
  • 3The scale factor is the ratio between corresponding sides of similar figures
  • 4You can find missing sides in similar figures by using proportions or the scale factor
  • 5Symbol for similarity: ∼\sim (e.g., △ABC∼△DEF\triangle ABC \sim \triangle DEF)

Frequently Asked Questions

Congruent figures are exactly the same size AND shape (scale factor = 1). Similar figures have the same shape but can be different sizes. All congruent figures are similar, but not all similar figures are congruent.
Congruent figures are exactly the same size AND shape (scale factor = 1). Similar figures have the same shape but can be different sizes. All congruent figures are similar, but not all similar figures are congruent.
Yes! When the second figure is smaller than the first, the scale factor is less than 1. For example, if figure B is half the size of figure A, the scale factor from A to B is 0.50.5 or 12\frac{1}{2}.
Yes! All squares have four right angles (90°) and four equal sides. Since the angles match and the sides are proportional (all multiplied by the same scale factor), all squares are similar to each other.

Glossary

Similar figures
Figures that have the same shape but not necessarily the same size
Corresponding angles
Angles in the same position in similar figures
Corresponding sides
Sides in the same position in similar figures
Scale factor
The ratio between corresponding sides of similar figures
Proportional
Having the same ratio
Today's challengeOne problem, three tries

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Congruent Triangles

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AA Similarity

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ASA and AAS Congruence

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