Similar Figures

Learn what makes figures similar and how to use proportions to find missing measurements.

Intermediate25 minLesson
Printable worksheet

Definition

Two figures are similar if they have the same shape but not necessarily the same size. Similar figures have:
  1. 1.Corresponding angles that are equal
  2. 2.Corresponding sides that are proportional
We use the symbol ∼\sim to show similarity. If triangle ABC is similar to triangle DEF, we write:
△ABC∼△DEF\triangle ABC \sim \triangle DEF
The scale factor is the ratio between corresponding sides of similar figures.

Try it now

Triangle ABC has sides 2, 3, and 4 cm. Triangle DEF has sides 4, 6, and 8 cm. What is the scale factor from ABC to DEF?

Worked Examples

Triangle ABC has sides 3 cm, 4 cm, and 5 cm. Triangle DEF has sides 6 cm, 8 cm, and 10 cm. Are these triangles similar?
1

Check if side ratios are equal

63=2\frac{6}{3} = 2, 84=2\frac{8}{4} = 2, 105=2\frac{10}{5} = 2 → All ratios equal 2
2

Verify the scale factor

Each side of DEF is exactly 2 times the corresponding side of ABC → Scale factor = 2
3

Conclude

Since all corresponding sides have the same ratio, 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 have the same shape but can be different sizes.

Correct: Similar = same shape, different size allowed. Congruent = same shape AND same size.

Not matching corresponding sides correctly

Why it's wrong: When comparing similar figures, you must match sides in the correct order based on their positions.

Correct: Always match the shortest side to shortest, longest to longest, or use angle positions to identify corresponding sides.

Adding instead of multiplying by the scale factor

Why it's wrong: Scale factor is a multiplicative relationship, not additive.

Correct: If scale factor is 2, multiply the original length by 2, don't add 2 to it.

Interactive Visual

Similar triangles

534ABC1068DEF△ABC ∼ △DEF

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

  • DE : AB = 10 : 5 = 2
  • EF : BC = 6 : 3 = 2
  • FD : CA = 8 : 4 = 2

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

Function graph

0510152025300510xyA12B8C12D8E18F12G18H12

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
Two figures are similar if they have the same _____ but not necessarily the same _____.

Why It Matters

Similar figures appear everywhere in our world:
  • Maps and blueprints: A map is similar to the actual land it represents
  • Photography: Enlarging or reducing photos maintains similarity
  • Architecture: Models of buildings are similar to the real structures
  • Art: Artists use similarity to create perspective and depth
  • Engineering: Scaled prototypes help test designs before full-scale production
Understanding similarity helps us work with objects of different sizes while maintaining the same proportions.

Real World Applications

Map Reading

Maps use similarity to represent large areas on small paper. The scale tells you the ratio between map distance and actual distance.

Example:

If a map has a scale of 1:50,000, then 1 cm on the map represents 50,000 cm (or 500 m) in real life.

1Try It Yourself

On a map with scale 1:25,000, two cities are 8 cm apart.

What is the actual distance between the cities in kilometers?

Step 1: Write the mathematical expression

Calculate: 8×250008 \times 25000 cm, then convert to km

Photo Enlargement

When you enlarge or shrink a photo, you create a similar figure. The aspect ratio stays the same.

Example:

A 4 by 6 inch photo enlarged by a factor of 2 becomes 8 by 12 inches. The ratio 4:6 = 8:12 = 2:3.

2Try It Yourself

A photo is 10 cm wide and 15 cm tall. You want to enlarge it to 24 cm wide.

How tall will the enlarged photo be?

Step 1: Write the mathematical expression

Set up proportion: 1015=24x\frac{10}{15} = \frac{24}{x}

Key Takeaways

  • 1Similar figures have the same shape but not necessarily the same size
  • 2Corresponding angles in similar figures are equal
  • 3Corresponding sides in similar figures are proportional
  • 4The scale factor is the ratio between corresponding sides
  • 5Use proportions to find missing measurements in similar figures

Frequently Asked Questions

Yes! All squares have four 90-degree angles and four equal sides. Any two squares will have matching angles and proportional sides, making them similar.
Yes! All squares have four 90-degree angles and four equal sides. Any two squares will have matching angles and proportional sides, making them similar.
No. While all rectangles have four 90-degree angles, their side ratios can differ. A 2 by 4 rectangle is not similar to a 2 by 6 rectangle because the ratios 2:4 and 2:6 are not equal.
Look at the position of sides relative to angles. The shortest side of one figure corresponds to the shortest side of the other. You can also use matching angle names (e.g., side AB corresponds to side DE).

Glossary

Similar figures
Figures that have the same shape but not necessarily the same size
Corresponding sides
Sides in similar figures that are in the same relative position
Corresponding angles
Angles in similar figures that are in the same relative position
Scale factor
The ratio of corresponding sides between two similar figures
Proportion
An equation stating that two ratios are equal

Formula Card

Scale Factor

k=side of larger figurecorresponding side of smaller figurek = \frac{\text{side of larger figure}}{\text{corresponding side of smaller figure}}

The ratio between corresponding sides of similar figures

Proportion

ab=cd\frac{a}{b} = \frac{c}{d}

An equation showing that two ratios are equal

Cross-Multiplication

a×d=b×ca \times d = b \times c

If two fractions are equal, their cross products are equal

Today's challengeOne problem, three tries

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