Similar shapes

Here you will learn about similar shapes, including what they are and how to identify them.

Students will first learn about similar shapes as part of geometry in 8 th grade.

What are similar shapes?

Similar shapes have sides of different lengths, but all corresponding sides are related by the same scale factor. All the corresponding angles in the similar shapes are equal and the corresponding lengths are in the same ratio.

For example, these two rectangles are similar shapes because:

Similar shapes 1 US

  • The scale factor from shape A to shape B is 2.

Similar shapes 2 US

  • The ratio of the bases is 2\text{:}4 which simplifies to 1\text{:}2.

Similar shapes 3 US

  • The ratio of the heights is also 1\text{:}2.

NOTE: Since the opposite sides of a rectangle are congruent, by comparing the base and the height, you have compared all sides of the rectangle.

  • The angles are all 90^{\circ}.

Similar shapes 4 US

What are similar shapes?

What are similar shapes?

Common Core State Standards

How does this relate to 8 th grade math?

  • Grade 8: Geometry (8.G.A.2)
    Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

[FREE] 2D Shape Worksheet (Grade 2 to 4)

[FREE] 2D Shape Worksheet (Grade 2 to 4)

[FREE] 2D Shape Worksheet (Grade 2 to 4)

Use this quiz to check your grade 2 to 4 students’ understanding of 2D shape. 15+ questions with answers covering a range of 2nd, 3rd and 4th grade 2D shape topics to identify areas of strength and support!

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[FREE] 2D Shape Worksheet (Grade 2 to 4)

[FREE] 2D Shape Worksheet (Grade 2 to 4)

[FREE] 2D Shape Worksheet (Grade 2 to 4)

Use this quiz to check your grade 2 to 4 students’ understanding of 2D shape. 15+ questions with answers covering a range of 2nd, 3rd and 4th grade 2D shape topics to identify areas of strength and support!

DOWNLOAD FREE

How to decide if shapes are similar

In order to decide if shapes are similar:

  1. Decide which sides are pairs of corresponding sides.
  2. Find the ratios of the sides.
  3. Check if the ratios are the same.

Similar shapes examples

Example 1: identifying similar rectangles

Are these shapes similar?

Similar shapes 5 US

  1. Decide which sides are pairs of corresponding sides.

The bases of the rectangles are a pair of corresponding sides.

The heights of the rectangles are a pair of corresponding sides.

2Find the ratios of the sides.

When writing the ratios, the order is very important.

Here the ratio is \text{length }A\text{:}\text{ length }B .

The ratio of the bases is 1\text{:}2.

The ratio of the heights is 2\text{:}4, which simplifies to 1\text{:}2.

3Check if the ratios are the same.

The ratios for the corresponding lengths are the same.

The scale factor from shape A to shape B is 2.

The rectangles are similar shapes.

Example 2: identifying similar triangles

Are these shapes similar?

Similar shapes 6 US

Decide which sides are pairs of corresponding sides.

Find the ratios of the sides.

Check if the ratios are the same.

How to find a missing length

In order to find a missing side in a pair of similar shapes:

  1. Decide which sides are pairs of corresponding sides.
  2. Find the scale factor.
  3. Use the scale factor to find the missing length.

Missing length examples

Example 3: finding a missing length on a quadrilateral

Here are two similar shapes. Find the length QR.

Similar shapes 7 US

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

Example 4: finding a missing length

Here are two similar triangles. Find the length BC.

Similar shapes 10 US

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

How to find a missing length in a triangle

In order to find a missing side in a pair of triangles when you are not told that the triangles are similar:

  1. Use angle facts to determine which angles are equal.
  2. Redraw the triangles side by side.
  3. Decide which sides are pairs of corresponding sides.
  4. Find the scale factor.
  5. Use the scale factor to find the missing length.

Missing length in a triangle examples

Example 5: finding a missing length in a triangle

Find the value of x.

Similar shapes 13 US

Use angle facts to determine which angles are equal.

Redraw the triangles side by side.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

Example 6: finding a missing length in a triangle

Find the value of x.

Similar shapes 18 US

Use angle facts to determine which angles are equal.

Redraw the triangles side by side.

Decide which sides are pairs of corresponding sides.

Find the scale factor.

Use the scale factor to find the missing length.

Teaching tips for similar shapes

  • Introduce the topic with familiar, regular polygons, before progressing to more complex or irregular shapes.

  • Use a variety of activities to teach students this topic, such as worksheets, interactive websites, or manipulatives. You can even challenge students to create similar shapes with given materials, having them justify their conclusions.

Easy mistakes to make

  • Confusing the order of ratios
    Make sure that you are consistent with your ratios. For example,
    In this example, always write the A value first, and then the B value.
    1\text{:}3 and 2\text{:}6. The ratios are equal, so these shapes are similar shapes.

    Similar shapes 23 US

  • Using only what the shapes look like to decide if they are similar
    Often diagrams for questions involving similar figures are NOT drawn to scale.
    So, use the measurements given, rather than just looking or trying to measure on your own.

  • Forgetting that shapes can be similar but in different orientations
    The second shape may be in a different orientation to the first shape. The shapes can still be similar. For example, here shape A and Shape B are similar. Shape B is an enlargement of shape A by scale factor 2.

    Similar shapes 24 US
    Here, shape B has been rotated to make the similarity easier to see.

    Similar shapes 25 US

  • Thinking scale factor is always a whole number
    The scale factor can also be a fraction or a decimal. When the fraction or decimal is less than 1, it means the shape shrank.

Practice similar shapes questions

1. Consider if these shapes are similar:

 

Similar shapes 26 US

Yes – sides in ratio 1\text{:}4

GCSE Quiz False

No – sides in ratio 1\text{:}3 and 1\text{:}4

GCSE Quiz False

No – sides in ratio 1\text{:}3 and 1\text{:}2

GCSE Quiz False

Yes – sides in ratio 1\text{:}3

GCSE Quiz True

The shapes are similar, since the ratios of the corresponding sides are the same.

 

Similar shapes 27 US

 

The ratio of the bases is \hspace{0.75cm} 3\text{:}9

 

which simplifies to \hspace{1.45cm} 1\text{:}3

 

and the ratio of the heights is \hspace{0.15cm} 1\text{:}3

2. Consider if these shapes are similar:

 

Similar shapes 28 US

Yes – sides in ratio 2\text{:}1

GCSE Quiz True

Yes – sides in ratio 3\text{:}1

GCSE Quiz False

No – sides in ratio 2\text{:}1 and 1\text{:}2

GCSE Quiz False

No – sides in ratio 3\text{:}1 and 2\text{:}1

GCSE Quiz False

Comparing the base and height as shown creates the ratios 8\text{:}2 and 4\text{:}4, which are not equal.

 

Similar shapes 29 US

 

However if you rotate the shapes, they are similar since the ratios of the corresponding sides are the same.

 

Similar shapes 29 US-1

 

The ratio of the height is \hspace{0.65cm} 4\text{:}2

 

which simplifies to \hspace{1.45cm} 2\text{:}1

 

and the ratio of the base is \hspace{0.4cm} 8\text{:}4

 

which simplifies to \hspace{1.45cm} 2\text{:}1

3. These shapes are similar. Find the value of x.

 

Similar shapes 31 US

x=11 
GCSE Quiz False

x=10
GCSE Quiz True

x=9
GCSE Quiz False

x=13
GCSE Quiz False

The ratio of the bases is \hspace{0.75cm} 6\text{:}12

 

which simplifies to \hspace{1.45cm} 1\text{:}2

 

The scale factor from the red parallelogram to the blue is 2.

 

x=2 \times 5=10

4. These shapes are similar. Find the value of x.

 

Similar shapes 32 US

x=11 
GCSE Quiz False

x=10
GCSE Quiz False

x=12
GCSE Quiz True

x=12.5
GCSE Quiz False

The ratio of the bases is \hspace{0.75cm} 6\text{:}9

 

which simplifies to \hspace{1.45cm} 1\text{:}\cfrac{9}{6}

 

or \hspace{3.65cm} 1\text{:}1.5

 

The scale factor from the small shape to the larger is 1.5.

 

x=8\times{1.5}=12

5. Find the value of x.

 

Similar shapes 33 US

x=11 
GCSE Quiz False

x=13
GCSE Quiz False

x=16
GCSE Quiz False

x=9
GCSE Quiz True

Use the parallel lines to identify equal angles.

 

Similar shapes 34 US

 

Then find pairs of corresponding sides.

 

Similar shapes 35 US

 

The ratio of the corresponding sides is 4\text{:}12, which simplifies to 1\text{:}3.

 

The scale factor from the smaller triangle to the larger is 3.

 

x=3\times{3}=9

6. Find the value of x.

 

Similar shapes 36 US

x=8
GCSE Quiz True

x=9
GCSE Quiz False

x=10
GCSE Quiz False

x=11
GCSE Quiz False

Use the parallel lines to identify equal angles.

 

Similar shapes 37 US

 

Then find pairs of corresponding sides.

 

Similar shapes 38 US

 

The ratio of the corresponding sides is 9\text{:}6, which simplifies to 1\text{:}23.

 

The scale factor from the larger triangle to the smaller is \cfrac{2}{3}.

 

x=12\times\cfrac{2}{3}=8

Similar shapes FAQs

Can similar shapes be different sizes?

Yes, similar shapes have different side lengths, but the ratio between corresponding side lengths is always equal.

What is a dilation?

A dilation is when a shape’s size is increased by a scale factor. A dilation creates a similar shape.

See also: Dilations

What is the Pythagorean theorem?

The theorem states that the sides of a right triangle are related by a^2+b^2=c^2, where a and b are the sides that form the right angle and c is the hypotenuse.

See also: Pythagorean Theorem

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