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Symmetry​

What is symmetry?

  • Symmetry in mathematics can refer to one of two types
    • Line (or Plane) symmetry which deals with reflections and mirror images of shapes or parts of shapes in both 2D and 3D
    • Rotational symmetry which deals with how often a shape looks identical (congruent) when it has been rotated

Rotational Symmetry

What is (the order of) rotational symmetry?

  • Rotational symmetry refers to the number of times a shape looks the same as it is rotated 360° about its centre
  • This number is called the order of rotational symmetry
  • Tracing paper can help work out the order of rotational symmetry
    • Draw an arrow on the tracing paper so you can easily tell when you have turned it through 360°

 

Symm Notes fig3a (1), downloadable IGCSE & GCSE Maths revision notes

Symm Notes fig3a (2), downloadable IGCSE & GCSE Maths revision notesSymm Notes fig3b, downloadable IGCSE & GCSE Maths revision notes

 

  • Notice that returning to the original shape contributes 1 to the order
    • This means a shape can never have order 0
    • A shape with rotational symmetry order 1 may be described as not having any rotational symmetry

      (The only time it looks the same is when you get back to the start)

Exam Tip

  • Tracing paper may help for rotational symmetry
    • One trick is to draw an arrow facing upwards so that when you rotate the tracing paper you know when it is back to its original position

Worked example

For the shape below, shade exactly 4 more squares so that the shape has rotational symmetry order 4.

For the shape below, shade exactly 4 more squares so that the shape has rotational symmetry order 4.

3-1-line-and-rotation-symmetry-we

The shape below appears the same 4 times if rotated through 360 degrees

 

3-1-1-rotation-symmetry-we-answer

Lines of Symmetry

What is line symmetry?

  • Line symmetry refers to shapes that can have mirror lines added to them
    • Each side of the line of symmetry is a reflection of the other side
  • Lines of symmetry can be thought of as a folding line too
    • Folding a shape along a line of symmetry results in the two parts sitting exactly on top of each other

Symm Notes fig2a, downloadable IGCSE & GCSE Maths revision notes

  • It can help to look at shapes from different angles – turn the page to do this

Symm Notes fig2b, downloadable IGCSE & GCSE Maths revision notes

 

  • Some questions will provide a shape and a line of symmetry
    • In these cases you need to complete the shape
  • Be careful with diagonal lines of symmetry
    • Use tracing paper to trace the shape and the reflection line and then flip on the line to see how the shape will reflect
  • “Two–way” reflections occur if the line of symmetry passes through the shape

Symm Notes fig2c (1), downloadable IGCSE & GCSE Maths revision notes Symm Notes fig2c (2), downloadable IGCSE & GCSE Maths revision notes

Symm Notes fig1, downloadable IGCSE & GCSE Maths revision notes

 

How do I solve problems involving symmetry?

  • Symmetry can be used to help solve missing length and angle problems

Symm Notes fig4 (1), downloadable IGCSE & GCSE Maths revision notes Symm Notes fig4 (2), downloadable IGCSE & GCSE Maths revision notes

Exam Tip

  • It may help to draw a diagram and add lines of symmetry to it or add to a diagram if one is given in a question
  • You should be provided with tracing paper in the exam, use this to help you

Worked example

For the shape below,

3-1-line-and-rotation-symmetry-we

 

(a)
Write down the number of lines of symmetry.
  
The only line of symmetry is shown below.

3-1-1-line-symmetry-we-answer

Answer = 1

 

(b)
Shade exactly 4 more squares so that the shape has 4 lines of symmetry.
     
The shape below has a horizontal, a vertical, and 2 diagonal lines of symmetry.

  

3-1-1-rotation-symmetry-we-answer

Planes of Symmetry

 

What is a plane of symmetry?

  • A plane is a flat surface that can be any 2D shape
  • A plane of symmetry is a plane that splits a 3D shape into two congruent (identical) halves
  • If a 3D shape has a plane of symmetry, it has reflection symmetry
    • The two congruent halves are identical, mirror images of each other
  • All prisms have at least one plane of symmetry
    • Cubes have 9 planes of symmetry
    • Cuboids have 3 planes of symmetry
    • Cylinders have an infinite number of planes of symmetry
    • The number of planes of symmetry in other prisms will be equal to the number of lines of symmetry in its cross-section plus 1
  • Pyramids can have planes of symmetry too
    • The number of planes of symmetry in other pyramids will be equal to the number of lines of symmetry in its 2D base
    • If the base of the pyramid is a regular polygon of n sides, it will have n planes of symmetry

3-1-1-cie-igcse-planes-of-symmetry-diagram-1

Exam Tip

If you’re unsure in the exam, consider the properties of the 3D shape.

  • Is it a prism or a pyramid?
  • How many lines of symmetry are there in the 2D faces or cross-section?

Worked example

The diagram below shows a cuboid of length 8 cm, width 5 cm and height 11 cm.

Write down the number of planes of symmetry of this cuboid.
 

cie-igcse-2020-oct-nov-p4-tz3-q6a

A plane of symmetry is where a shape can be “sliced” such that it is symmetrical.
A cuboid with three different pairs of opposite rectangles has 3 planes of symmetry.

3 planes of symmetry

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