Similar Figures Examples in Math

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

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

Concept Recap

Similar figures have the same shape with corresponding angles equal and corresponding sides proportional.

One figure is an enlarged or reduced copy of another—same shape, same angles, but possibly 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: Similar figures have proportional corresponding sides and equal corresponding angles—shape is preserved, size is not.

Common stuck point: Students match non-corresponding sides when writing proportions.

Sense of Study hint: Mark matching angles first, then pair sides opposite those angles.

Worked Examples

Example 1

easy
Two similar triangles have corresponding sides 6 cm and 10 cm. If the smaller triangle has a perimeter of 21 cm, find the perimeter of the larger triangle.

Solution

  1. 1
    Scale factor (larger to smaller): k = \dfrac{10}{6} = \dfrac{5}{3}.
  2. 2
    Perimeters of similar figures scale by the same ratio as corresponding sides.
  3. 3
    Perimeter of larger triangle = 21 \times \dfrac{5}{3} = 35 cm.

Answer

The larger triangle has perimeter 35 cm.
All linear measurements in similar figures scale by the same ratio k. Only areas scale by k^2 and volumes by k^3. Once the scale factor is identified from one pair of corresponding sides, all other proportions follow.

Example 2

hard
Rectangle ABCD \sim Rectangle EFGH with AB = 8, BC = 5, EF = 12. Find FG and the ratio of areas.

Practice Problems

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

Example 1

easy
Triangle PQR \sim Triangle STU. PQ = 4, ST = 6, QR = 7. Find TU.

Example 2

medium
Two similar cones have base radii 3 cm and 7 cm. What is the ratio of their volumes?
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Background Knowledge

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

similarityproportionsscale drawings