Similarity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Similarity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Two figures are similar if they have the same shape but possibly different sizes.

A photo and its enlargement are similar—same shape, different size.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Similarity preserves angles and ratios, but not actual lengths.

Common stuck point: Students confuse similar with congruent. Similar shapes have the same shape but can differ in size. All circles are similar; not all rectangles are.

Sense of Study hint: Compare the ratios of corresponding sides. If all the ratios are equal, the shapes are similar even if the sizes differ.

Worked Examples

Example 1

medium
Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, AC = 10, and DE = 9, find EF and DF.

Solution

  1. 1
    Find the scale factor: k = \frac{DE}{AB} = \frac{9}{6} = 1.5.
  2. 2
    Multiply each corresponding side by the scale factor: EF = BC \times 1.5 = 8 \times 1.5 = 12.
  3. 3
    DF = AC \times 1.5 = 10 \times 1.5 = 15.

Answer

EF = 12, \quad DF = 15
Similar figures have equal corresponding angles and proportional corresponding sides. The ratio between any pair of corresponding sides is constant (the scale factor).

Example 2

hard
A tree casts a 15 m shadow at the same time a 2 m pole casts a 3 m shadow. How tall is the tree?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Two similar rectangles have widths of 4 cm and 10 cm. If the smaller rectangle has a length of 6 cm, find the length of the larger rectangle.

Example 2

medium
Two similar triangles have corresponding side lengths in the ratio 3:5. If the perimeter of the smaller triangle is 27 cm, find the perimeter of the larger triangle.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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