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, meaning all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).

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

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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: Two figures are similar when all matching angles are equal and all matching sides share one scale factor.

Common stuck point: The procedure for similarity is the easy part; the trap is expecting similar figures to have equal sides. Asking "Are all corresponding angles equal and all corresponding sides in the same ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are all corresponding angles equal and all corresponding sides in the same ratio?

Worked Examples

Example 1

medium
Triangle ABCABC is similar to triangle DEFDEF. If AB=6AB = 6, BC=8BC = 8, AC=10AC = 10, and DE=9DE = 9, find EFEF and DFDF.

Answer

EF=12,DF=15EF = 12, \quad DF = 15

First step

1
Find the scale factor: k=DEAB=96=1.5k = \frac{DE}{AB} = \frac{9}{6} = 1.5.

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Example 2

hard
A tree casts a 1515 m shadow at the same time a 22 m pole casts a 33 m shadow. How tall is the tree?

Example 3

medium
In △ABC\triangle ABC, a line through DD on ABAB parallel to BCBC meets ACAC at EE. If AD=3,DB=6AD=3, DB=6, AE=4AE=4, find ECEC.

Example 4

hard
In △ABC\triangle ABC, the altitude from the right angle to the hypotenuse has length hh. Legs are a=6,b=8a=6, b=8, hypotenuse c=10c=10. Find hh.

Example 5

challenge
Triangles ABCABC and XYZXYZ are similar. The longest side of ABCABC is 1414 and the longest side of XYZXYZ is 2121. The area of ABCABC is 4040. Find the area of XYZXYZ.

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 44 cm and 1010 cm. If the smaller rectangle has a length of 66 cm, find the length of the larger rectangle.

Example 2

medium
Two similar triangles have corresponding side lengths in the ratio 3:53:5. If the perimeter of the smaller triangle is 2727 cm, find the perimeter of the larger triangle.

Example 3

easy
Two similar triangles have a scale factor of 33. A side of the small one is 44. Find the matching side of the large one.

Example 4

easy
Two similar figures have corresponding sides 55 and 1515. What is the scale factor (large to small)?

Example 5

easy
Are all squares similar to each other?

Example 6

easy
In two similar triangles, corresponding angles are related how?

Example 7

easy
A 66-foot person casts a 44-foot shadow. A tree casts a 2020-foot shadow at the same time. Set up the proportion for the tree's height hh.

Example 8

easy
Two triangles each have angles 50∘50^\circ and 60∘60^\circ. Are they similar?

Example 9

easy
Two similar rectangles have a scale factor of 22. The small one has area 1010. Find the large one's area.

Example 10

easy
True or false: two figures can be congruent but not similar.

Example 11

medium
Triangle ABC∼ABC \sim triangle DEFDEF. AB=8AB = 8, DE=12DE = 12, and BC=10BC = 10. Find EFEF.

Example 12

medium
In triangle ABCABC, a line parallel to BCBC cuts ABAB at DD and ACAC at EE. If AD=4AD = 4, DB=6DB = 6, and AE=6AE = 6, find ECEC.

Example 13

medium
Two similar solids have a scale factor of 22. How do their volumes compare?

Example 14

medium
A map has scale 1:500001 : 50000. Two towns are 44 cm apart on the map. Find the real distance in km.

Example 15

medium
Two similar triangles have areas 99 and 2525. Find the ratio of their corresponding sides.

Example 16

medium
A photo 44 in by 66 in is enlarged so its longer side becomes 1515 in. If the enlargement is similar, find the new shorter side.

Example 17

medium
In a right triangle, the altitude to the hypotenuse creates two smaller triangles. How do they relate to the original?

Example 18

medium
Triangle AA has sides 3,4,53, 4, 5. Triangle BB has sides 9,12,169, 12, 16. Are they similar?

Example 19

challenge
Two similar triangles have areas in the ratio 4:94:9. The perimeter of the smaller is 2424. Find the perimeter of the larger.

Example 20

challenge
A cone is filled with water to half its height. What fraction of the cone's total volume is the water?

Example 21

challenge
In triangle ABCABC, point DD on ABAB and EE on ACAC make △ADE∼△ABC\triangle ADE \sim \triangle ABC with DE∥BCDE \parallel BC. If AD=xAD = x, AB=x+6AB = x + 6, and the area of △ADE\triangle ADE is one-quarter the area of △ABC\triangle ABC, find xx.

Example 22

challenge
Explain why all circles are similar to each other, and why this means π\pi is the same for every circle.

Example 23

easy
Two similar triangles have scale factor 44. A side on the small triangle is 77. Find the matching side on the large triangle.

Example 24

easy
Two similar rectangles have sides 3×53 \times 5 and 9×k9 \times k. Find kk.

Example 25

easy
A 55-ft post casts a 33-ft shadow. At the same time a flagpole casts a 2424-ft shadow. Find the flagpole's height.

Example 26

easy
A map uses scale 1:250001:25000. Two parks are 66 cm apart on the map. Find the real distance in km.

Example 27

medium
Triangles ABC∼DEFABC\sim DEF with AB=9,AC=12,BC=15AB=9, AC=12, BC=15. If DE=6DE=6, find EFEF.

Example 28

medium
Two similar triangles have areas 3636 and 100100. Find the ratio of their corresponding sides.

Example 29

medium
In a right triangle with legs 33 and 44, the altitude to the hypotenuse divides the triangle into two smaller right triangles. Are all three triangles similar?

Example 30

medium
Two similar solids have scale factor 33. The smaller has volume 4040 cm3^3. Find the larger's volume.

Example 31

medium
In two similar triangles, one has sides 5,12,135, 12, 13 and the other has its shortest side 2020. Find the longer leg of the second triangle.

Example 32

medium
Triangles ABC∼XYZABC\sim XYZ with ∠A=∠X\angle A=\angle X, ∠B=∠Y\angle B=\angle Y. If AB=10,XY=15AB=10, XY=15, XZ=12XZ=12, find ACAC.

Example 33

hard
In △ABC\triangle ABC, DD lies on ABAB with AD:DB=2:3AD:DB=2:3, and DE∥BCDE \parallel BC meets ACAC at EE. Find the ratio of the area of △ADE\triangle ADE to △ABC\triangle ABC.

Example 34

hard
Two similar cones have volumes 125125 and 343343 cm3^3. Find the ratio of their slant heights.

Example 35

hard
A scale model of a building has scale 1:2001:200. The real building has a roof area of 40004000 m2^2. Find the model's roof area in cm2^2.

Example 36

hard
△ADE∼△ABC\triangle ADE \sim \triangle ABC with DE∥BCDE \parallel BC. AD=xAD=x, AB=12AB=12. If area of △ADE\triangle ADE equals 1/91/9 of △ABC\triangle ABC, find xx.

Example 37

challenge
A cone of height HH is filled with water to height hh from the apex. Find the fraction of the cone's volume that is filled, in terms of hh and HH.
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Background Knowledge

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

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