Planes of Symmetry in 3D Shapes

Find planes of symmetry of 3D solids such as cuboids, cubes, prisms and pyramids by picturing a flat cut into mirror halves.

In the exam for Cambridge O Level Maths 4024, Cambridge IGCSE Maths 0580 · 5 real past-paper questions below

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A Flat Cut Into Mirror Halves

A plane of symmetry is a flat cut through a solid. The two pieces are mirror images of each other.

It is the 3D version of a line of symmetry: a line splits a flat shape, a plane splits a solid.

Rule

Test a cut by asking: if the cut were a mirror, would one half look exactly like the other half's reflection?

Cuboids and Cubes

The 1-mark question: how many planes of symmetry has this cuboid? Cut halfway across each pair of opposite faces.

Worked example

1
$3 pairs of opposite faces$ — A cuboid has 3 pairs of opposite faces: top and bottom, front and back, left and right.
2
$cuboid: 3 middle cuts$ — One cut halfway between the two faces of each pair gives two mirror halves. A cuboid with all edges different has no other planes.
3
$cube: 3 + 6 = 9$ — A cube also has square faces, so a diagonal cut through a pair of opposite edges gives mirror halves too. There are 6 of those on top of the 3 middle cuts.

Common mistake

Counting the diagonal cuts for a cuboid. Its faces are not squares, so a diagonal cut gives two halves that do not mirror each other.

Prisms, Pyramids and Round Solids

For a prism, look at its end. Each line of symmetry of the end gives a cut along the length. Add one more cut across the middle.

Worked example

1
$equilateral triangle end: 3 lines of symmetry$ — Look at the end of the prism. An equilateral triangle has 3 lines of symmetry.
2
$3 cuts along the length$ — Each line of symmetry of the end gives one plane, standing along the length of the prism.
3
$+ 1 cut halfway along$ — A cut halfway along the length, parallel to the ends, gives two mirror halves too. There is always exactly one of these.
4
$3 + 1 = 4 planes$ — Add the two kinds of cut.

Solid	Planes
Cuboid, all edges different	3
Cube	9
Equilateral triangular prism	4
Square-based pyramid	4
Cylinder, cone, sphere	infinitely many

Key idea

A pyramid has no cut across the middle, because the top and bottom are different. Its planes all pass through the point on top.

Square Ends, Pyramids and the Axis

Two solids the exam loves: a cuboid with a square end, and a pyramid. The prism rule counts the first. The base counts the second.

Worked example

1
$4 cm \times 4 cm \times 9 cm$ — Two edges are equal, so two faces are squares. This is a prism with a square end.
2
$square end: 4 lines of symmetry$ — A square has 4 lines of symmetry: 2 through the middles of opposite sides and 2 along the diagonals. Each gives a cut along the length.
3
$+ 1 cut halfway along$ — The cut across the middle of the length, as for any prism.
4
$4 + 1 = 5 planes$ — More than the 3 of a cuboid with all edges different, but fewer than the 9 of a cube.

Now a pyramid whose base is an equilateral triangle and whose apex is directly above the centre of the base.

Worked example

1
$base: 3 lines of symmetry$ — Look at the base only. An equilateral triangle has 3 lines of symmetry.
2
$each line \to 1 plane up to the apex$ — A line of symmetry of the base, together with the apex above the centre, makes one plane.
3
$no cut across the middle$ — Cutting parallel to the base leaves a small pyramid on top and a different shape below, so they are not mirror images.
4
$3 planes$ — A pyramid has as many planes as its base has lines of symmetry.

Key idea

A pyramid also has an axis of rotational symmetry. It is the straight line from the apex down to the centre of the base. In a diagram, name it with the two letters at its ends.

Which Method? Read the Question

Picture the solid as a mirror would split it.

You see	You do
Cuboid	one cut per pair of faces: 3 (5 if one pair of faces is square)
Cube	3 middle cuts + 6 diagonal: 9
Prism	lines of symmetry of the end + 1
Pyramid	lines of symmetry of the base (apex above the centre)
A flat shape	Lines of Symmetry

Key idea

Round solids like a cylinder have infinitely many planes: any cut through the middle along the axis works.

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