Cambridge IGCSE Mathematics 0580: Symmetry – Study Guide

This study guide teaches subtopic 4.5, Symmetry, from the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027. Line symmetry and rotational symmetry in two dimensions (C4.5 and E4.5 part 1) are in both tiers. The symmetry of prisms, cylinders, pyramids and cones (E4.5 part 2) is Extended only, so it is examined only on Paper 2 (non-calculator) and Paper 4 (calculator).

Use it with the 0580 course hub and the printable 0580 checklist. The wider topic is summarised in the geometry study guide and the geometry revision notes, and you can test it with the geometry practice questions. To find gaps across the course, try the Core diagnostic or the Extended diagnostic.

What this subtopic covers

0580 reference	What you must be able to do	Tier
C4.5, E4.5.1	Recognise line symmetry in two dimensions	Core and Extended
C4.5, E4.5.1	Recognise the order of rotational symmetry in two dimensions	Core and Extended
C4.5, E4.5.1 (notes)	Know the properties of triangles, quadrilaterals and polygons that come directly from their symmetries	Core and Extended
E4.5.2	Recognise symmetry properties of prisms, cylinders, pyramids and cones, e.g. identify planes and axes of symmetry	Extended only

Line symmetry

A line of symmetry (mirror line) splits a shape into two halves that are mirror images: fold along it and one half fits exactly on the other. To test a line, go from each vertex straight across the line at 90° and the same distance beyond it. If there is a matching vertex there every time, it is a line of symmetry.

A diagonal is not automatically a line of symmetry. A rectangle’s diagonal cuts it into two congruent triangles, but folding along it does not make them fit.

Rotational symmetry

A shape has rotational symmetry if it looks exactly the same after a turn of less than 360° about a fixed point (the centre of rotation). The order of rotational symmetry is the number of positions in one full turn in which the shape looks the same.

• Every shape fits onto itself after a full 360° turn, so the smallest order is 1. Order 1 means no rotational symmetry.
• If the order is n, the smallest turn that maps the shape onto itself is 360° ÷ n.
• Tracing paper helps: turn a tracing once round the centre and count the fits.

A useful link between the two

If a flat shape has at least one line of symmetry, its order of rotational symmetry equals its number of lines. A shape with no lines can still have rotational symmetry: a parallelogram has order 2.

Symmetry of triangles and quadrilaterals

Learn these counts and why the properties below follow from them.

Shape	Lines of symmetry	Order of rotational symmetry
Equilateral triangle	3	3
Isosceles triangle (not equilateral)	1	1
Scalene triangle	0	1
Square	4	4
Rectangle (not a square)	2	2
Rhombus (not a square)	2	2
Parallelogram (not a rectangle or rhombus)	0	2
Kite (not a rhombus)	1	1
Isosceles trapezium	1	1
Trapezium (not isosceles)	0	1

Properties that come from these symmetries:

• Isosceles triangle: the line of symmetry joins the apex to the midpoint of the base. It bisects the apex angle, meets the base at 90° and reflects one base angle onto the other, so the base angles are equal.
• Rectangle: the two lines of symmetry join the midpoints of opposite sides. They are not the diagonals.
• Rhombus: the two lines of symmetry are the diagonals. So each diagonal bisects the angles it passes through, and the diagonals cross at 90°.
• Square: it has the lines of a rectangle and the lines of a rhombus, four in all.
• Parallelogram: a half-turn about the point where the diagonals cross maps it onto itself. So opposite sides are equal, opposite angles are equal and the diagonals bisect each other.
• Isosceles trapezium: its line of symmetry joins the midpoints of the parallel sides, so the base angles are equal in pairs and the diagonals are equal.

Worked example 1 (non-calculator): a rhombus

PQRS is a rhombus. Angle PQR = 116°. Find angle QPR.

The diagonals of a rhombus are its lines of symmetry.
QS is a line of symmetry, so it bisects angle PQR:
  angle PQS = 116° ÷ 2 = 58°
Let the diagonals meet at M. The diagonals of a rhombus cross at 90°,
so angle PMQ = 90°.
In triangle PMQ:  angle QPM = 180° − 90° − 58° = 32°
M lies on PR, so angle QPR = 32°.

Check: PR bisects angle QPS, so angle QPS = 64°, and 116° + 64° = 180° (co-interior angles).

Worked example 2 (non-calculator): an isosceles trapezium

ABCD is an isosceles trapezium with AB parallel to DC and AD = BC. Angle DAB = 68°. Find angles ABC, ADC and BCD.

The line of symmetry joins the midpoints of AB and DC.
It reflects A onto B and D onto C.
  angle ABC = angle DAB = 68°
AB is parallel to DC, so co-interior angles add to 180°:
  angle ADC = 180° − 68° = 112°
  angle BCD = angle ADC = 112°   (by the symmetry)
Check: 68 + 68 + 112 + 112 = 360°

Symmetry of regular polygons

A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

• If n is odd, every line of symmetry goes through a vertex and the midpoint of the opposite side (a regular pentagon has 5 such lines).
• If n is even, half the lines go through pairs of opposite vertices and half go through the midpoints of pairs of opposite sides (a regular octagon has 4 of each).

The smallest turn that maps a regular polygon onto itself is 360° ÷ n. That is also the size of one exterior angle, which links this subtopic to the polygon angle facts in C4.6.

Worked example 3 (non-calculator): working back from a turn

A regular polygon fits onto itself after a turn of 24°, and 24° is the smallest turn that does this. Find the number of sides, the number of lines of symmetry and the size of one interior angle.

Order of rotational symmetry = 360° ÷ 24° = 15
A regular polygon has order n, so n = 15 sides.
Lines of symmetry = 15
Exterior angle = 360° ÷ 15 = 24°
Interior angle = 180° − 24° = 156°

Symmetry on a grid

You may be asked to draw a line of symmetry on a grid, write its equation, or add points so a shape has a given symmetry.

Worked example 4 (Extended): the equation of a line of symmetry

Triangle PQR has vertices P(1, 4), Q(4, 1) and R(5, 5). Show that the triangle is isosceles and find the equation of its line of symmetry.

PR² = (5 − 1)² + (5 − 4)² = 16 + 1 = 17
QR² = (5 − 4)² + (5 − 1)² = 1 + 16 = 17
PQ² = (4 − 1)² + (1 − 4)² = 9 + 9 = 18
PR = QR = √17, so the triangle is isosceles with apex R.
Swapping the x- and y-coordinates maps P(1, 4) to Q(4, 1)
and maps R(5, 5) to itself.
Reflection in y = x swaps x and y, so the line of symmetry is y = x.

It has 1 line of symmetry and order 1. Equations of lines are covered in coordinate geometry.

Worked example 5: completing a shape with rotational symmetry

Points A(1, 1), B(4, 1) and C(5, 3) are three vertices of a hexagon. The hexagon has rotational symmetry of order 2 about the point (3, 3). Find the other three vertices.

Order 2 means a half-turn (180°) about (3, 3) maps the shape onto itself.
A half-turn about (3, 3) sends (x, y) to (6 − x, 6 − y),
because (3, 3) is the midpoint of each point and its image.
  A(1, 1) → (5, 5)
  B(4, 1) → (2, 5)
  C(5, 3) → (1, 3)
The hexagon is A(1, 1), B(4, 1), C(5, 3), (5, 5), (2, 5), (1, 3).

This hexagon has order 2 but no lines of symmetry, like a parallelogram. Rotations are described fully in the transformations and vectors study guide.

Symmetry of solids (Extended only)

The syllabus example for E4.5 part 2 is identifying planes and axes of symmetry.

• A plane of symmetry cuts a solid into two halves that are mirror images. It is the 3D version of a line of symmetry.
• An axis of symmetry is a line through the solid about which it can be turned and look the same in more than one position in a full turn. The order counts those positions.

Right prisms

For a right prism, look at the cross-section first.

• Each line of symmetry of the cross-section gives a plane of symmetry that runs along the length of the prism.
• There is one more plane: the plane halfway along the length, parallel to the two ends.
• The axis along the length, through the centre of the ends, has the same order as the cross-section’s rotational symmetry.

Solid	Planes of symmetry	Rotational symmetry about the axis through the ends
Cuboid, all edges different (e.g. 2 × 3 × 5)	3	order 2
Cuboid with two square faces (e.g. 3 × 3 × 7)	5	order 4
Cube	9	order 4
Triangular prism, equilateral cross-section	4	order 3
Triangular prism, isosceles cross-section	2	none (order 1)
Regular hexagonal prism	7	order 6
Cylinder	infinitely many	any angle of turn

A cube is the exception to the cross-section rule: it is a prism in three directions, so it has 3 + 6 = 9 planes. A cuboid with all edges different has three axes of symmetry, each of order 2, through the centres of pairs of opposite faces. A cylinder’s planes are every plane containing its central axis, plus the plane halfway up.

Right pyramids and cones

In a right pyramid the apex is directly above the centre of the base. Every plane of symmetry contains the vertical axis through the apex (a regular tetrahedron is an exception). So:

• planes of symmetry = lines of symmetry of the base
• order about the vertical axis = order of rotational symmetry of the base
• there is no horizontal plane of symmetry.

Solid	Planes of symmetry	Order about the vertical axis
Square-based right pyramid	4	4
Rectangle-based right pyramid (base not square)	2	2
Regular hexagonal-based right pyramid	6	6
Right circular cone	infinitely many	any angle of turn

Worked example 6: naming a plane of symmetry

ABCDEFGH is a cuboid. The base ABCD is a 4 cm by 4 cm square and the edges AE, BF, CG and DH are vertical and 9 cm long. Is the plane ACGE a plane of symmetry?

Plane ACGE contains the vertical edges AE and CG and the base diagonal AC.
A diagonal of a square is a line of symmetry of the square.
Reflecting in plane ACGE maps B to D, F to H and the cuboid onto itself.
Plane ACGE is a plane of symmetry.

If the base were a 4 cm by 6 cm rectangle, AC would not be a line of symmetry of the base, so ACGE would not be a plane of symmetry.

Common errors

• Counting the diagonals of a rectangle or parallelogram as lines of symmetry.
• Giving the order of rotational symmetry as 0. The smallest order is 1, which means no rotational symmetry.
• Saying a parallelogram has 2 lines of symmetry because it has order 2.
• Counting each line twice on an even-sided regular polygon: a regular hexagon has 6 lines, not 12.
• Adding a horizontal plane of symmetry to a pyramid or cone.
• For a prism, forgetting the plane halfway along the length, or counting it twice.

Next steps

Revise with the geometry revision notes, then try the geometry practice questions. Extended candidates can go on to the circle theorems II study guide, which uses the symmetry of chords and tangents (E4.8). The course hub lists other resources for syllabus 0580.

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Subject content, section 4 Geometry, C4.5 Symmetry (Core subject content) and E4.5 Symmetry (Extended subject content).