Similar - Definition, Examples, Quiz, FAQ, Trivia
Learn how shapes can be alike but different sizes, and how to work with scale factors
What are Similar Figures?
Similar figures are shapes that have the same shape but may be different sizes. They look identical except that one is a scaled-up or scaled-down version of the other.
Think of it like looking at the same picture on your phone and on a movie screen - they show the same thing, just at different sizes. The key is that all corresponding angles are equal, and the sides are proportional.
Similar figures are everywhere in our world! When you see a model car that looks just like a real car but smaller, those are similar figures. When you see a small picture and a big poster of the same image, they are similar.
Key Concept
Similar figures have the same shape but different sizes. Their corresponding angles are equal and sides are proportional.
Properties of Similar Figures
Similar figures have two important properties:
1. Corresponding angles are equal: This means that all matching angles in the similar shapes have the same measurement.
2. Corresponding sides are proportional: This means that if one side is twice as long as its matching side in the other figure, then all sides will be twice as long. This ratio is called the scale factor.
For example, if two triangles are similar:
- Angle A in first triangle = Angle D in second triangle
- Angle B = Angle E
- Angle C = Angle F
- Side AB / Side DE = Side BC / Side EF = Side AC / Side DF
Similar vs. Congruent
Congruent figures have the same size and shape. Similar figures have the same shape but different sizes. All congruent figures are similar, but not all similar figures are congruent.
Scale Factor
The scale factor tells us how much larger or smaller one similar figure is compared to another. It's the ratio of any two corresponding lengths in the two figures.
Scale Factor Formula
If a model car is 1/24 the size of a real car, the scale factor is 1/24. This means:
- 1 cm on the model = 24 cm on the real car
- The real car is 24 times larger than the model
Scale factor affects area and volume too:
- If lengths are multiplied by k (scale factor)
- Areas are multiplied by k²
- Volumes are multiplied by k³
Original Size
Scale Factor 2
Scale Factor 3
Similar Triangles
Triangles are the most common similar figures we work with in geometry. There are special rules that help us determine if triangles are similar:
AAA (Angle-Angle-Angle) Rule: If all three angles of one triangle are equal to all three angles of another triangle, the triangles are similar.
SSS (Side-Side-Side) Rule: If all three sides of one triangle are proportional to the three sides of another triangle, the triangles are similar.
SAS (Side-Angle-Side) Rule: If two sides are proportional and the included angle is equal, the triangles are similar.
Example: If Triangle ABC has angles 40°, 60°, 80° and Triangle DEF has angles 40°, 60°, 80°, then the triangles are similar by AAA rule.
Real-World Application
Surveyors use similar triangles to measure distances to faraway objects. By creating a small similar triangle, they can calculate distances they couldn't directly measure.
Area and Volume of Similar Figures
When we enlarge a shape using a scale factor, the area and volume don't increase by the same amount as the sides. Here's how it works:
Area: When all dimensions are multiplied by a scale factor k, the area is multiplied by k².
Example: If you double the dimensions (k=2), the area becomes 4 times larger (2²=4).
Volume: When all dimensions are multiplied by a scale factor k, the volume is multiplied by k³.
Example: If you triple the dimensions (k=3), the volume becomes 27 times larger (3³=27).
This is why giant versions of small objects would be much heavier than you might expect. A giant soda can that's twice as tall and twice as wide would actually hold 8 times as much soda!
| Scale Factor | Side Length | Area | Volume |
|---|---|---|---|
| 1 | Original | Original | Original |
| 2 | 2× | 4× | 8× |
| 3 | 3× | 9× | 27× |
| 0.5 | ½× | ¼× | ⅛× |
Similar Figures Practice Quiz
Test your understanding with this 5-question quiz. Choose the correct answer for each question.
Frequently Asked Questions
Here are answers to common questions about similar figures:
Geometry Trivia
Discover interesting facts about similar figures and geometry:
Ancient Origins
The concept of similar figures dates back to ancient Egypt, where surveyors used similar triangles to measure land after Nile floods. The Greek mathematician Thales first formalized the principles around 600 BC.
Fractal Geometry
Fractals are special mathematical shapes that exhibit self-similarity - they look similar at any scale. Zooming into a fractal reveals patterns similar to the whole shape, just like similar figures.
Space Exploration
NASA engineers use similarity principles when designing scale models of spacecraft for wind tunnel testing. The models are mathematically similar to the actual spacecraft to accurately predict flight characteristics.
Art and Perspective
Renaissance artists used similarity principles to create realistic perspective in paintings. Objects farther away are drawn as smaller but similar versions of closer objects, creating the illusion of depth.
