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 are scaled copies: corresponding angles match exactly and corresponding sides share one constant ratio.

Common stuck point: The procedure for similar figures is the easy part; the trap is treating similar as congruent. Asking "Do the two figures have all corresponding angles equal and all corresponding sides in one common ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the two figures have all corresponding angles equal and all corresponding sides in one common ratio?

Worked Examples

Example 1

easy
Two similar triangles have corresponding sides 66 cm and 1010 cm. If the smaller triangle has a perimeter of 2121 cm, find the perimeter of the larger triangle.

Answer

The larger triangle has perimeter 3535 cm.

First step

1
Scale factor (larger to smaller): k=106=53k = \dfrac{10}{6} = \dfrac{5}{3}.

Full solution

  1. 2
    Perimeters of similar figures scale by the same ratio as corresponding sides.
  2. 3
    Perimeter of larger triangle =21×53=35= 21 \times \dfrac{5}{3} = 35 cm.
All linear measurements in similar figures scale by the same ratio kk. Only areas scale by k2k^2 and volumes by k3k^3. Once the scale factor is identified from one pair of corresponding sides, all other proportions follow.

Example 2

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

Example 3

medium
Two similar polygons have perimeters 3636 and 6060. The smaller has a side of length 99. Find the matching side on the larger.

Example 4

medium
Two similar triangles have areas 3232 and 5050. The smaller has a side 1212. Find the matching side on the larger.

Example 5

hard
A small statue is similar to a big statue. The big one is 33 times as tall and weighs 135135 kg. Estimate the small statue's weight assuming same density.

Example 6

challenge
Two similar prisms have volumes in ratio 27:6427:64 and total surface areas aa and bb. If a=108a=108 cm2^2, find bb.

Practice Problems

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

Example 1

easy
Triangle PQR∼PQR \sim Triangle STUSTU. PQ=4PQ = 4, ST=6ST = 6, QR=7QR = 7. Find TUTU.

Example 2

medium
Two similar cones have base radii 33 cm and 77 cm. What is the ratio of their volumes?

Example 3

easy
Two similar triangles have corresponding sides 44 and 1010. Find the scale factor (large to small).

Example 4

easy
Two similar pentagons have scale factor 44. If a side on the small one is 33, find the matching side on the large one.

Example 5

easy
Two similar triangles have side ratio 3:53:5. Find the perimeter ratio.

Example 6

easy
Two similar squares have side ratio 1:61:6. Find the area ratio.

Example 7

medium
Triangle ABC∼ABC\sim Triangle DEFDEF with AB=5AB=5, DE=20DE=20. If AC=8AC=8, find DFDF.

Example 8

medium
Two similar rectangles have areas 4848 and 108108. Find the linear scale factor (large to small).

Example 9

medium
Two similar pyramids have heights 66 cm and 99 cm. Find the ratio of their surface areas.

Example 10

medium
Two similar cylinders have radii 33 and 55. Find the ratio of their lateral surface areas.

Example 11

medium
A photograph is enlarged so that the new perimeter is 33 times the old. By what factor does the area grow?

Example 12

hard
Two similar solids have surface areas 4848 and 108108. Find the ratio of their volumes.

Example 13

hard
A scale model of a building uses scale 1:501:50. The real building has 2,5002{,}500 m2^2 of glass. How much glass does the model need?

Example 14

hard
Two similar cones have volumes 54π54\pi and 128π128\pi. Find the ratio of their heights.

Example 15

hard
Triangle ABCABC has AB=6,BC=8,CA=10AB=6, BC=8, CA=10. A similar triangle has its longest side equal to 2525. Find the perimeter of the similar triangle.

Example 16

challenge
A model airplane (scale 1:481:48) has wings totaling 0.50.5 m2^2 in area. Find the real airplane's wing area in m2^2.
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Background Knowledge

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

similarityproportionsscale drawings