Rotations

Here we will learn about rotations about a point, including how to describe rotations.

There are also rotations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are rotations?

Rotations are transformations that turn a shape around a fixed point.

To rotate a shape we need:

  • a centre of rotation
  • an angle of rotation (given in degrees)
  • a direction of rotation – either clockwise or anti-clockwise. (Anti-clockwise direction is sometimes known as counterclockwise direction).

Rotations Image 1

E.g.

Rotate shape A 90^o clockwise, about a fixed point.

Rotations Image 2

Shape A has been rotated a quarter turn clockwise to give shape B.

E.g.

Rotate shape A 180^o about a fixed point.

Rotations Image 3

The shape A has been rotated a half turn to give shape B.

Whether the direction is clockwise or anti-clockwise (counterclockwise) is irrelevant. 

Using tracing paper can be very useful when using rotations.

We call the original shape the object and the rotated shape the image

For rotations the object shape and the image shape are congruent because they are the same shape and the same size.

As the lengths of the shape have been kept the same so the shapes are said to have isometry.

See also: Rotational symmetry

What are rotations?

What are rotations?

How to rotate a shape about a fixed point

In order to rotate a shape about a fixed point:

  1. Trace the shape.
  2. Rotate the tracing paper about the centre of enlargement.
  3. Draw the rotated shape onto the grid.

Explain how to rotate a shape about a fixed point

Explain how to rotate a shape about a fixed point

Rotations worksheet

Rotations worksheet

Rotations worksheet

Get your free rotations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE
x
Rotations worksheet

Rotations worksheet

Rotations worksheet

Get your free rotations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Rotations examples

Example 1: rotate a shape about a fixed point

Rotate the shaded shape 90^o clockwise about the fixed point:

Rotations Example 1

  1. Trace the shape.

Use a pencil and trace the shape onto a piece of tracing paper.

Rotations Example 1 Step 1 NEW

2Rotate the tracing paper about the centre of enlargement.

Use the pencil and put the tip onto the fixed point. Pivot the tracing paper a quarter turn clockwise.

Rotations Example 1 Step 2

3Draw the rotated shape onto the grid.

Carefully lift the tracing paper and draw the rotated shape in the correct position.

Rotations Example 1 Step 3

Note: one of the vertices of the triangle has not moved. This is also known as an invariant point of the shape.

Example 2: rotate a shape about a fixed point

Rotate the shaded shape 180^o about the fixed point:

Rotations Example 2

Use a pencil and trace the shape onto a piece of tracing paper.


Rotations Example 2 Step 1

Use the pencil and put the tip onto the fixed point. Pivot the tracing paper.


Rotations Example 2 Step 2

Carefully lift the tracing paper and draw the rotated shape in the correct position.


Rotations Example 2 Step 3


Note: one of the vertices of the triangle has not moved. This is also known as an invariant point of the shape.

Example 3: rotate a shape about a centre of rotation

Rotate the shaded shape 90^o anti-clockwise about (3,3)

v

On the diagram mark the centre of rotation.


Rotations Example 3 Step 1

It can be useful to add a line on the diagram extending from the shape to the centre of rotation.  Use a pencil and trace the shape onto a piece of tracing paper.


Rotations Example 3 Step 2

Use the pencil and put the tip onto the centre of rotation. Pivot the tracing paper. It may be useful to add a line connecting the shape and the centre of rotation.


Rotations Example 3 Step 3 NEW

Carefully lift the tracing paper and draw the rotated shape in the correct position.


Those dotted lines are extra, but they help to show more clearly that the shape has been rotated correctly.


Rotations Example 3 Step 4

Example 4: rotate a shape about a centre of rotation

Rotate the shaded shape 180^o about O:

Rotations Example 4

On the diagram mark the centre of rotation. O stands for the Origin of the coordinate grid and has the coordinates (0,0)


Rotations Example 4 Step 1

It can be useful to add a line on the diagram extending from the shape to the centre of rotation.  Use a pencil and trace the shape onto a piece of tracing paper.


Rotations Example 4 Step 2 NEW

Use the pencil and put the tip onto the centre of rotation. Pivot the tracing paper. It may be useful to add a line connecting the shape and the centre of rotation.


Rotations Example 4 Step 3 NEW

Carefully lift the tracing paper and draw the rotated shape in the correct position.


Those dotted lines are extra, but they help to show more clearly that the shape has been rotated correctly.


Rotations Example 4 Step 4

How to describe a rotation

In order to describe a rotation:

  1. Trace the shape.
  2. Rotate the tracing paper.
  3. Write down the description.

Describing rotations examples

Example 5: describe a rotation

Describe the rotation of shape A to shape B

Rotations Example 5

Use a pencil and trace the object shape onto a piece of tracing paper.


Rotations Example 5 Step 1 NEW

Have a think about where the centre of rotation might be. Use the pencil and put the tip onto that point.  Pivot the tracing paper to check. It may take a few tries until you find the correct centre of rotation.


Rotations Example 5 Step 2 NEW

Make sure you state that it is a rotation. Then give the angle of rotation and if necessary the direction of rotation. Also give the coordinates of the centre of rotation.


Rotation 180^o about the point (0,1)


Since the rotation is a half-turn, no direction is needed.


Since the rotation was 180^o, if we connect the vertices in corresponding pairs, these lines all cross at the centre of rotation.


Rotations Example 5 Step 3

Example 6: describe a rotation

Describe the rotation of shape A to shape B

Rotations Example 6

Use a pencil and trace the object shape onto a piece of tracing paper.


Rotations Example 6 Step 1 NEW

Have a think about where the centre of rotation might be.  Use the pencil and put the tip onto that point.  Pivot the tracing paper to check.  It may take a few tries until you find the correct centre of rotation.


Rotations Example 6 Step 2 NEW

Make sure you state that it is a rotation.  Then give the angle of rotation and if necessary the direction of rotation.  Also give the coordinates of the centre of rotation.


Rotation 90^o clockwise about the point (-1,-1)

Common misconceptions

  • Rotate with the correct angle

Adding a small arrow onto your tracing paper can help to rotate the shape by the correct angle.

Rotations Common Misconceptions Image 1 NEW

  • Clockwise and anticlockwise

You may need to look at a clock to remind yourself of the difference between clockwise and anticlockwise.

Rotations Common Misconceptions Image 2

  • The origin

The origin of a coordinate grid has the coordinates (0,0) .  It is commonly denoted as O.  It is used often as the centre of enlargement.

  • Position of the centre of rotation

The centre of rotation can be within the object shape.

E.g.

Rotations Common Misconceptions Image 3

  • Alternative angles and directions

A rotation of 270^o clockwise is a correct alternative to 90^o anti-clockwise.

A rotation of 270^o anti-clockwise is a correct alternative to 90^o clockwise.

Practice rotation questions

1. Rotate the shaded shape 180^o about the centre of rotation:

 

Rotations Practice Question 1

Rotations Practice Question 1 Answer 2

GCSE Quiz False

Rotations Practice Question 1 Answer 3

GCSE Quiz False

Rotations Practice Question 1 Answer 4

GCSE Quiz False

Rotations Practice Question 1 Answer 1

GCSE Quiz True

The object shape has to be rotated a half-turn. It needs to have been rotated about the centre of rotation. It can not have been reflected.

2. Rotate the shaded shape 90^o anti-clockwise about the centre of rotation:

 

Rotations Practice Question 2

Rotations Practice Question 2 Answer 1

GCSE Quiz True

Rotations Practice Question 2 Answer 2

GCSE Quiz False

Rotations Practice Question 2 Answer 3

GCSE Quiz False

Rotations Practice Question 2 Answer 4

GCSE Quiz False

The object shape has to be rotated 90^o anticlockwise. The centre of rotation should be used. The additional extra dotted lines may help to make this rotation clearer.

 

Rotations Practice Question 2 Explanation Image

3. Rotate the shaded shape 90^o clockwise about (0,0):

 

Rotations Practice Question 3

Rotations Practice Question 3 Answer 2

GCSE Quiz False

Rotations Practice Question 3 Answer 1

GCSE Quiz True

Rotations Practice Question 3 Answer 3

GCSE Quiz False

Rotations Practice Question 3 Answer 4

GCSE Quiz False

The object shape has to be rotated 90^o clockwise. The centre of rotation should be the Origin. The additional extra dotted lines help to make this rotation clearer.

 

Rotations Practice Question 3 Explanation Image

4. Rotate the shaded shape ​ 180^o ​ about ​ (-1,0):

 

Rotations Practice Question 4

Rotations Practice Question 4 Answer 2

GCSE Quiz False

Rotations Practice Question 4 Answer 3

GCSE Quiz False

Rotations Practice Question 4 Answer 1

GCSE Quiz True

Rotations Practice Question 4 Answer 4

GCSE Quiz False

The centre of rotation should be the (-1,0). The additional extra dotted lines help to make this rotation clearer.

 

Rotations Practice Question 4 Explanation Image

5. Describe the rotation of shape A to shape B

 

Rotations Practice Question 5

Rotation
90^o clockwise
About (1,1)

GCSE Quiz False

Rotation
90^o clockwise
About the origin

GCSE Quiz True

Rotation
90^o anticlockwise
About the origin

GCSE Quiz False

Rotation
90^o anticlockwise
About (1,1)

GCSE Quiz False

Make sure you know which is the original, object shape and which is the image shape. The additional extra dotted lines help to make this rotation clearer. The centre of rotation is (0,0), the origin.

 

Be careful with the direction of the rotation, or you may give the inverse. The inverse would give the rotation of the image to the object.

 

Rotations Practice Question 5 Answer Image

6. Describe the rotation of shape A to shape B

 

Rotations Practice Question 6

Rotation
180^o
About (1,1)

GCSE Quiz False

Rotation
90^o anticlockwise
About the origin

GCSE Quiz False

Reflection
180^o
About (0,0)

GCSE Quiz False

Rotation
180^o
About (-1,-1)

GCSE Quiz True

Make sure you know which is the original, object shape and which is the image shape. Since this is a half-turn the direction of the 180^o is not needed.

 

The additional extra dotted lines help to make this rotation clearer. The centre of rotation is (-1,-1).

 

Rotations Practice Question 6 Explanation Image

Rotations GCSE questions

1. On the grid, rotate the kite 90^o clockwise about the point O.

 

Rotations GCSE Question 1 Image 1

 

(2 marks)

Show answer

Rotations GCSE Question 1 Image 2

 

For any rotation of 90^o

(1)

 

For a rotation of 90^o and in the correct direction and about the correct centre of rotation

(1)

2. Rotate the triangle 90^o anti-clockwise about (0,0)

 

Rotations GCSE Question 2 Image 1

 

(2 marks)

Show answer

Rotations GCSE Question 2 Image 2

 

For any rotation of 90^o

(1)

 

For a rotation of 90^o but in the correct direction and about the correct centre of rotation

(1)

3. Describe fully the single transformation which maps triangle A onto triangle B.

 

Rotations GCSE Question 3

 

(3 marks)

Show answer

Rotation

For stating the transformation is a rotation

(1)

 

180^o

For giving the angle, the direction is not needed

(1)

 

About the point (3,4) but in the correct direction and about the correct centre of rotation

(1)

Learning checklist

You have now learned how to:

  • Rotate 2D shapes on a grid
  • Rotate 2D shapes on a coordinate grid given a centre of rotation
  • Describe fully a rotation of a 2D shape

Beyond GCSE

Beyond the scope of GCSE mathematics is three-dimensional rotation, also known as 3D rotation which includes quaternion rotation.  Euler angles are used in 3D rotation.  We can also use matrices to help us with more complex rotations; one of these is known as a rotation matrix.

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths revision programme.

x

GCSE Maths Scheme of Work Guide

An essential guide for all SLT and subject leaders looking to plan and build a new scheme of work, including how to decide what to teach and for how long

Explores tried and tested approaches and includes examples and templates you can use immediately in your school.

Download free