Similarity of Geometric Figures

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The similarity between geometric figures is met when they have angles of the same size respectively and there is also proportionality between the sides of such figures. 

In an intuitive way, just as it happens with triangles, two similar figures are, in fact, an enlargement of the other.

Similarity of geometric figures image

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Test yourself on similarity of polygons!

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Look at the two similar rectangles below and calculate the perimeter of the larger rectangle.

141414XXX3.53.53.51.51.51.5

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Similarity of Geometric Figures

Example 1

Let's demonstrate this topic with an example.

We have an illustration of two similar rectangles, ABCD ABCD and KLMN KLMN .

In both rectangles all angles are right angles (equivalent to 90º 90º ).

Moreover, each side of the large rectangle KLMN KLMN is greater than the respective side in the small rectangle ABCD ABCD .

That is, KL=12 KL=12 in the large rectangle KLMN KLMN is twice as long as AB=6 AB=6 in the small rectangle ABCD ABCD , and KN=8 KN=8 in the large rectangle KLMN KLMN is twice as long as AB=4 AB=4 in the small rectangle ABCD ABCD .

Example 2

These two squares are similar:

Similarity of geometric figures image

The two corresponding angles are equal since all angles are right angles. The ratio between the corresponding sides, that is, the scale factor is
2:1 2:1

or, in other words, each side of the larger square measures twice as much as each side of the small square

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Question 1

1027.51.5The two parallelograms above are similar. The ratio between their sides is 3:4.

What is the ratio between the the areas of the parallelograms?

Question 2

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

Question 3

In front of you are two hexagons with a similarity ratio. Which angles are corresponding?

4.54.54.50.750.750.753334443336663330.50.50.52222.662.662.66222444AAABBBCCCDDDEEEFFFMMMNNNOOOPPPRRRJJJ

Example 3 - Similar Figures

The two pentagons in the illustration are similar, meaning, the corresponding angles are equal. The ratio of similarity is 

EFAB=32=1.51 \frac{EF}{AB}=\frac{3}{2}=\frac{1.5}{1}

The two pentagons in the illustration are similar

That is, the length of each side in the pentagon FGHIJ FGHIJ is 1.5 1.5 times greater than that of its corresponding side in the pentagon ABCDE ABCDE

If you are interested in this article, you might also be interested in the following articles

  • Similarity of triangles and polygons
  • Ratio of similarity
  • Similar triangles
  • Criterion of similarity between two triangles

In the blog of Tutorela you will find a wide variety of mathematics articles

Examples and exercises with solutions on similarity of geometric figures

examples.example_title

Look at the two similar rectangles below and calculate the perimeter of the larger rectangle.

141414XXX3.53.53.51.51.51.5

examples.explanation_title

Let's remember that in a rectangle there are two pairs of parallel and equal sides.

We will call the small triangle 1 and the large triangle 2.

We calculate the perimeter of the small triangle:

P1=2×3.5+2×1.5=10 P_1=2\times3.5+2\times1.5=10 Since we know that the rectangles are similar:

3.514=p1p2 \frac{3.5}{14}=\frac{p_1}{p_2}

We place the data we know for the perimeter:

3.514=10p2 \frac{3.5}{14}=\frac{10}{p_2}

3.514×p2=10 \frac{3.5}{14}\times p_{_2}=10

p2=10×143.5 p_2=10\times\frac{14}{3.5}

P2=40 P_2=40

examples.solution_title

40 cm

examples.example_title

1027.51.5The two parallelograms above are similar. The ratio between their sides is 3:4.

What is the ratio between the the areas of the parallelograms?

examples.explanation_title

The square of the ratio between the sides is equal to the ratio between the areas of the parallelograms:

32:42=9:16 3^2:4^2=9:16

examples.solution_title

9:16

examples.example_title

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

examples.explanation_title

We try to verify the ratio of similarity.

We examine if:

ABEF=ACEG \frac{AB}{EF}=\frac{AC}{EG}

We replace the data:

710=36 \frac{7}{10}=\frac{3}{6}

710≠12 \frac{7}{10}\ne\frac{1}{2}

The ratio is not equal, so the rectangles are not similar.

examples.solution_title

Not similar

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Question 1

Choose the correct answer

8888881212126668886661414146661414143.53.53.51.51.51.53.53.53.561.5

Question 2

Which figure shows a pair of similar polygons?

Question 3

Select the correct answer

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