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Recognise Rotational Symmetry

In this worksheet, students will recall and identify the rotational symmetry of common 2D shapes, plus locate squares in a grid to colour to create rotational symmetry to a specified order.

'Recognise Rotational Symmetry' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Pearson Edexcel, Eduqas, OCR

Curriculum topic:   Geometry and Measures, Congruence and Similarity

Curriculum subtopic:   Properties and Constructions, Plane Isometric Transformations

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

What is rotational symmetry?

 

A shape has rotational symmetry when it still looks the same after some rotation of less than one full turn through 360°.

A shape's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same within one rotation.

 

 

 

Applied in geometric shapes

 

If we take a square and rotate it through a full turn, it will look exactly the same on four occasions: after a quarter turn, half turn, three-quarters of a turn and a full turn.

We call this rotational symmetry of order 4.

 

If we take a rectangle and rotate it through a full turn, it will look the same twice: after half a turn and a full turn.

We call this rotational symmetry of order 2.

 

If we take an isosceles triangle and rotate it through a full turn, it will look the same only once: after a full turn.

We call this rotational symmetry of order 1.

 

 

 

e.g. What order of rotational symmetry does this shape have?

 

An equilateral triangle

 

If we rotated this shape through a full turn, we would reach this same image on three occasions: 1/3 of a turn, 2/3 of a turn and a full turn.

 

So this shape has rotation symmetry of order 3.

 

 

 

e.g. Which squares would we have to shade for this shape to have rotational symmetry of order 2?

 

 

'Order 2' means it would look the same twice within one full rotation of 360°.

If we divide the full turn (360°) by this, we find that it looks the same every 180°.

 

If we rotated the shape 180°, the shaded trio of squares would be in the bottom left corner instead, like this:

 

 

So this version of the square will have rotational symmetry to the order of 2

 

 

 

e.g. Which squares would we have to shade for this shape to have rotation symmetry of order 4?

 

 

'Order 4' means it would look the same four times within a full rotation.

If we divide the full turn (360°) by this, we find that it looks the same every 90°.

 

If we rotated the shape 90°, the shaded pair of squares would be in the bottom right corner.

If we rotated the shape another 90°, the shaded pair of squares would be in the bottom left corner.

If we rotated the shape a final 90°, the shaded pair of squares would be in the top left corner.

If we colour all these squares, our final image will look like this:

 

 

 

 

In this activity, we will recall and identify the rotational symmetry of common 2D shapes, plus locate squares in a grid to colour to create rotational symmetry to a specified order. 

Read the sentence below, then type a single word in the gap to accurately complete it. 

What order of rotational symmetry does a square have?

1

2

3

4

What order of rotational symmetry does a rectangle have?

1

2

3

4

Triangles have different orders of rotational symmetry depending on what type of triangle they are.

 

Read each statement below, then type a single number in each gap to complete each. 

 

1

2

3

4

Match each shape on the left to its correct order of rotational symmetry on the right.

Column A

Column B

Square
3
Equilateral triangle
4
Regular hexagon
6
Regular decagon
10

What is the order of rotational symmetry of the shape below?

 

4 by 4 grid with 4 blue coloured squares

What is the order of rotational symmetry of the shape below?

 

4 by 4 grid with outline of blue squares and centre of red

Which of the squares (A - M) would we need to shade grey to create rotational symmetry of order 2?

 

4 by 4 grid with 3 shaded squares

 

(The letters have not been written on this grid so you need to imagine that the squares are just white; they have been provided for identification purposes only.)

A

B

C

D

E

F

G

H

I

J

K

L

M

Which of the squares (A - M) would we need to shade grey to create rotational symmetry of order 4?

 

4 by 4 grid with 3 grey shaded squares

 

(The letters have not been written on this grid so you need to imagine that the squares are just white; they have been provided for identification purposes only.)

A

B

C

D

E

F

G

H

I

J

K

L

M

What is the order of rotational symmetry of a regular polygon?

 

Read the sentence below, then type a single word in the gap to accurately complete it. 

A

B

C

D

E

F

G

H

I

J

K

L

M

  • Question 1

Read the sentence below, then type a single word in the gap to accurately complete it. 

CORRECT ANSWER
EDDIE SAYS
Its really important that we get this wording straight in our mind as this is the terminology we will need to recognise and use in our exams. If we rotate a shape through 360°, the number of times when it looks exactly the same will define its rotational symmetry. We call this the order of rotational symmetry.
  • Question 2

What order of rotational symmetry does a square have?

CORRECT ANSWER
4
EDDIE SAYS
A square is a regular shape. Regular shapes will always have the same order of rotational symmetry as their number of sides. So a square has rotational symmetry to the order of four as it will look exactly the same on four occasions when it is rotated through 360°.
  • Question 3

What order of rotational symmetry does a rectangle have?

CORRECT ANSWER
2
EDDIE SAYS
It is a really common error to say "Four" here. The key thing to note with a rectangle is that is is not a regular shape, so its order of rotational symmetry will not match its number of sides. Try it... Take a piece of paper and turn it 90°. Does it look the same as when you started? A rectangular piece of paper will only look the same twice when rotated through 360° so its order of rotational symmetry is 2.
  • Question 4

Triangles have different orders of rotational symmetry depending on what type of triangle they are.

 

Read each statement below, then type a single number in each gap to complete each. 

 

CORRECT ANSWER
EDDIE SAYS
The equilateral triangle is a regular shape. What do we know about the order of rotational symmetry for a regular shape? That's right, it will match the number of sides of the shape, so an equilateral triangle has an order of rotational symmetry of 3. An isosceles triangle has two sides which are the same, so it can only be positioned in the same way once in a single turn. A common error is to say that a scalene triangle has an order of 0. No shape can have a rotational order of 0, as all shapes will look exactly the same as they return to the start after a full turn. So the lowest possible number we will see for an order of rotational symmetry is 1. Therefore, the scalene triangle (with no equal sides) also has an order of rotational symmetry of 1.
  • Question 5

Match each shape on the left to its correct order of rotational symmetry on the right.

CORRECT ANSWER

Column A

Column B

Square
4
Equilateral triangle
3
Regular hexagon
6
Regular decagon
10
EDDIE SAYS
Did you spot that these are all regular shapes? Remember the fact that a regular shape will have the same order of rotational symmetry as it does number of sides. So the regular hexagon, with 6 sides, will have an order of rotational symmetry of 6 too. Can you apply your knowledge of regular shapes to find the other matches here?
  • Question 6

What is the order of rotational symmetry of the shape below?

 

4 by 4 grid with 4 blue coloured squares

CORRECT ANSWER
2
Two
EDDIE SAYS
If we turn this grid through a full rotation of 360°, how many times would it exactly like this? As the top right and bottom left corners are an exact match, we can find two orientations which match this view in a full rotation. One will be on the full turn and the other on the half turn. Can you imagine this grid rotating through 180°?
  • Question 7

What is the order of rotational symmetry of the shape below?

 

4 by 4 grid with outline of blue squares and centre of red

CORRECT ANSWER
4
Four
EDDIE SAYS
If we turn this grid through a full rotation of 360°, how many times would it look exactly like this? We could turn it 90° or a quarter turn and it would match, then another 90°, and a final 90°, plus the full turn. So that is four turns in total.
  • Question 8

Which of the squares (A - M) would we need to shade grey to create rotational symmetry of order 2?

 

4 by 4 grid with 3 shaded squares

 

(The letters have not been written on this grid so you need to imagine that the squares are just white; they have been provided for identification purposes only.)

CORRECT ANSWER
A
E
F
EDDIE SAYS
Remember that 'order 2' means it will look exactly the same twice in a full rotation. Once, if we rotate it through 180°, and the other at the full turn (shown here). Where would the three shaded squares be if we turn this grid 180°? They are in the bottom right corner originally so, once the grid has been rotated a half turn, they will be in the top left corner. Which letters accurately identify their location in the top left quadrant?
  • Question 9

Which of the squares (A - M) would we need to shade grey to create rotational symmetry of order 4?

 

4 by 4 grid with 3 grey shaded squares

 

(The letters have not been written on this grid so you need to imagine that the squares are just white; they have been provided for identification purposes only.)

CORRECT ANSWER
A
C
D
E
F
G
J
K
L
EDDIE SAYS
'Order 4' means it will look exactly the same four times in a full rotation. Once, if we rotate it through 90°, then 180°, then 270° and the final time at the full turn of 360° (shown here). Where would the three shaded squares be if we turn this grid 90° clockwise? They are in the bottom right corner originally so, once the grid has been rotated a quarter turn, they will be in the bottom left corner. Which letters accurately identify their location in the bottom left quadrant? Can you repeat this process for the half and three-quarters turn?
  • Question 10

What is the order of rotational symmetry of a regular polygon?

 

Read the sentence below, then type a single word in the gap to accurately complete it. 

CORRECT ANSWER
EDDIE SAYS
A polygon is the name given to any two-dimensional shape with straight-line sides (including triangles, quadrilaterals, etc.) The use of the word 'regular' means that all the sides and internal angles of a given shape are the same. What does this tell us about the order of rotational symmetry? Remember our key rule: For regular shapes, the order of rotational symmetry will always match the number of sides of the shape. Amazing! You can now recall and identify the rotational symmetry of common 2D shapes, plus locate squares in a grid to colour to create rotational symmetry to a specified order.
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