Symmetry (Cambridge O Level Maths)

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

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°

• 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)

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

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

• 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

How do I solve problems involving symmetry?

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

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