Planes of Symmetry (Edexcel GCSE Maths): Revision Note

Exam code: 1MA1

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 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

A cube has 9 planes of symmetry, a cuboid has 3 planes of symmetry, a square based pyramid has 4 planes of symmetry

Can a 3D shape have rotational symmetry?

  • 3D shapes are able to be rotated around different axes

    • Depending on which axis the shape is rotated around, 3D shapes can have rotational symmetry

  • Recall that rotational symmetry is how many times the shape looks the same (congruent) when rotated through 360 degrees

    • See the example of the triangular prism where the cross-section is an equilateral triangle

A triangular prism (where the cross-section is an equilateral triangle) looks the same, 3 times, as it is rotated 360 degrees about an axis through the centre of the triangular cross section. Therefore it has rotational symmetry order 3.
A square pyramid has rotational symmetry order 4 about the vertical axis, as its base is a square. A cylinder has infinite rotational symmetry about the vertical axis, as its cross section is a circle. This would also be true for a cone.

Examiner Tips and Tricks

  • 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.
 

A cuboid ABCDEFGH, with AB = 8 cm, BC = 5 cm and CG = 11 cm.

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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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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