Similarity of Polygons - Examples, Exercises and Solutions

Question Types:
Similarity of Polygons: Using ratios for calculationSimilarity of Polygons: Worded problemsSimilarity of Polygons: Calculate The Missing Side based on the formulaSimilarity of Polygons: Identifying and defining elements

Similarity of Geometric Figures

Similarity in geometry refers to the relationship between two shapes that have the same shape but may differ in size. Two figures are similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. This means one figure can be obtained by resizing the other, either by scaling up or scaling down, without changing the shape.

While similarity is most commonly associated with triangles, it can apply to almost any shape or figure.

Similar geometric have these key properties:

  1. They have angles of the same size respectively. In other words, all corresponding angles between similar figures are equal, preserving the overall shape.
  2. Proportionality between the sides of such figures - the ratios of corresponding side lengths are the same across similar 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

Detailed explanation

Suggested Topics to Practice in Advance

  1. Similar Triangles
  2. Triangle similarity criteria
  3. Similarity of Triangles and Polygons
  4. Similarity ratio

Practice Similarity of Polygons

Question 1

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

141414XXX3.53.53.51.51.51.5

Question 2

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 3

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

Question 4

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

Question 5

Which shapes are similar?

8888881212126668886661414146661414143.53.53.51.51.51.53.53.53.561.5

Examples with solutions for Similarity of Polygons

Exercise #1

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

141414XXX3.53.53.51.51.51.5

Video Solution

Step-by-Step Solution

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

Answer

40 cm

Exercise #2

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?

Video Solution

Step-by-Step Solution

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

Answer

9:16

Exercise #3

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

Video Solution

Step-by-Step Solution

We first need to verify the ratio of similarity.

We examine if:

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

To do this, we substitute our values in:

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

710≠12 \frac{7}{10}\ne\frac{1}{2}

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

Answer

No

Exercise #4

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

Video Solution

Answer

Angle C = Angle O

Exercise #5

Which shapes are similar?

8888881212126668886661414146661414143.53.53.51.51.51.53.53.53.561.5

Video Solution

Answer

The rectangles are similar.

Question 1

Which figure shows a pair of similar polygons?

Question 2

Which statement is true?

Question 3

The polygons below are similar.

Calculate X.

XXX333121212888121212888

Question 4

Calculate the length of AB given that the triangles below are similar.

5.25.25.2121212555131313AAABBBCCCDDDEEEHHH

Question 5

Two similar rectangles are shown below.

What is the height of rectangle number 1?

3.53.53.5XXX66614141412

Exercise #6

Which figure shows a pair of similar polygons?

Video Solution

Answer

333333333333555555555555

Exercise #7

Which statement is true?

Video Solution

Answer

It cannot be determined.

Exercise #8

The polygons below are similar.

Calculate X.

XXX333121212888121212888

Video Solution

Answer

2

Exercise #9

Calculate the length of AB given that the triangles below are similar.

5.25.25.2121212555131313AAABBBCCCDDDEEEHHH

Video Solution

Answer

4.8

Exercise #10

Two similar rectangles are shown below.

What is the height of rectangle number 1?

3.53.53.5XXX66614141412

Video Solution

Answer

1.5

Question 1

Calculate X given that the polygons shown below are similar.

202020101010252525444XXX555

Question 2

ABCD is similar to EFGH.

If the perimeter of ABCD is 54 m, then what is the perimeter of EFGH?

666444AAABBBDDDCCCEEEFFFHHHGGG

Question 3

The two polygons below are similar.

Calculate the missing sides of the polygon ABCD.

333333333333555

Question 4

The price of a square rug that is 2 meters long is $600.

What is the price of a square rug that has a side length of 3 meters?

222333

Question 5

Fatima receives $350 for her work on a garden that measures 10 meters by 10 meters.

How much will she receive for working on a garden measuring 20 meters by 20 meters?

101010101010202020202020

Exercise #11

Calculate X given that the polygons shown below are similar.

202020101010252525444XXX555

Video Solution

Answer

2

Exercise #12

ABCD is similar to EFGH.

If the perimeter of ABCD is 54 m, then what is the perimeter of EFGH?

666444AAABBBDDDCCCEEEFFFHHHGGG

Video Solution

Answer

36 m

Exercise #13

The two polygons below are similar.

Calculate the missing sides of the polygon ABCD.

333333333333555

Video Solution

Answer

5

Exercise #14

The price of a square rug that is 2 meters long is $600.

What is the price of a square rug that has a side length of 3 meters?

222333

Video Solution

Answer

$1350

Exercise #15

Fatima receives $350 for her work on a garden that measures 10 meters by 10 meters.

How much will she receive for working on a garden measuring 20 meters by 20 meters?

101010101010202020202020

Video Solution

Answer

$1400