Symmetry
Line symmetry and rotational symmetry of flat shapes, the symmetries of triangles, quadrilaterals and regular polygons, and how symmetry explains their properties. Extended: planes of symmetry and axes of rotational symmetry of prisms, cylinders, pyramids and cones.
What the syllabus says
C4.5 (Core)
- Recognise line symmetry and order of rotational symmetry in two dimensions.
Notes and examples
- Includes properties of triangles, quadrilaterals and polygons directly related to their symmetries.
E4.5 (Extended)
- 1 Recognise line symmetry and order of rotational symmetry in two dimensions.
- 2 Recognise symmetry properties of prisms, cylinders, pyramids and cones.
Notes and examples
- Includes properties of triangles, quadrilaterals and polygons directly related to their symmetries.
- e.g. identify planes and axes of symmetry.
Quoted from the Cambridge IGCSE Mathematics 0580 syllabus for 2025 to 2027.
Line symmetry
A flat shape has line symmetry if a straight line divides it into two halves that are mirror images of each other. The line is called a line of symmetry or mirror line.
The folding test. Imagine folding the shape along the line. If the two halves fit exactly on top of each other, it is a line of symmetry. Another way to check: every point on one side must have a partner the same distance away on the other side, with the line joining them crossing the mirror line at right angles.
Some shapes have one line of symmetry, some several, and some none at all. The arrow below has exactly one.
A common trap is the diagonal of a rectangle. It cuts the rectangle into two identical triangles, but they are not mirror images across the diagonal: fold along it and the corners stick out. Identical halves are not enough; they must be reflections of each other in the line.
Rotational symmetry
A shape has rotational symmetry if it fits onto itself exactly when it is turned through less than a full turn about a point, the centre of rotation.
The order of rotational symmetry is the number of times the shape fits onto itself in one full turn ().
- Every shape fits onto itself after a full turn, so the order is always at least 1.
- A shape with order 1 has no rotational symmetry. Do not write order 0.
- If the order is , the smallest turn that maps the shape onto itself is degrees.
How to find the order. Trace the shape (in your head, or on tracing paper), mark one corner, and turn it. Count how many positions it fits in before the marked corner gets back to the start.
Line symmetry and rotational symmetry are separate. A parallelogram has rotational symmetry of order 2 but no lines of symmetry. A kite has one line of symmetry but no rotational symmetry (order 1).
Triangles, quadrilaterals and polygons
The table below gives the symmetries you need to know. The syllabus also includes the properties of these shapes that come from their symmetry.
Symmetry explains the properties. A reflection or rotation does not change lengths or angles, so whatever the symmetry swaps must be equal.
- Isosceles triangle: the line of symmetry goes from the apex to the midpoint of the base. It swaps the two base angles, so they are equal; it bisects the apex angle; and it meets the base at , at its midpoint.
- Kite: the line of symmetry is the diagonal through the vertices where the equal sides meet. It swaps the other two vertices, so the kite has one pair of equal angles. It also bisects the other diagonal at right angles.
- Parallelogram: the half turn about the centre swaps opposite sides and opposite angles, so they are equal, and the diagonals bisect each other.
- Rhombus: both diagonals are lines of symmetry, so they bisect the angles and cross at right angles.
Regular polygons. A regular polygon with sides has lines of symmetry and rotational symmetry of order . If is odd, each line goes from a vertex to the midpoint of the opposite side. If is even, half the lines join opposite vertices and half join the midpoints of opposite sides. The smallest angle of rotation, , is the same as the exterior angle.
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 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 | 1 | 1 |
| Isosceles trapezium | 1 | 1 |
| Regular polygon with sides |
Learn this table: questions ask for these numbers directly, and use them to identify a shape.
Worked example 1
For each shape, write down the number of lines of symmetry and the order of rotational symmetry: (a) a rhombus that is not a square, (b) an isosceles trapezium, (c) a regular octagon.
(a) A rhombus: its two diagonals are lines of symmetry (the lines through the midpoints of opposite sides are not). It fits onto itself after a half turn. 2 lines, order 2.
(b) An isosceles trapezium: one line, through the midpoints of the two parallel sides. A half turn puts the long parallel side where the short one was, so there is no rotational symmetry. 1 line, order 1.
(c) A regular octagon: 8 lines, order 8. Its smallest angle of rotation is .
Answer: (a) 2, 2 (b) 1, 1 (c) 8, 8
Worked example 2
On the 4 by 4 grid, three squares are shaded: the top left corner square and the top two squares of the central 2 by 2 block. Shade as few extra squares as possible so that the pattern has rotational symmetry of order 4. How many lines of symmetry does the finished pattern have?
Order 4 means the pattern fits after every quarter turn about the centre of the grid. Turn the top left corner square through quarter turns: it visits all four corners, so shade all four corners.
The two top squares of the central block turn onto the other two squares of the block, so shade the whole central 2 by 2 block.
Check the lines of symmetry: the vertical and horizontal centre lines and both diagonals each map the pattern onto itself. That is 4 lines.
Answer: Shade the other three corners and the bottom two squares of the central block (5 more squares). The finished pattern has 4 lines of symmetry.
Check yourself
Write down the number of lines of symmetry and the order of rotational symmetry of a regular pentagon.
Answer it in the portal, where your answer is checked as you type →
Check yourself
A quadrilateral has rotational symmetry of order 2 and exactly two lines of symmetry. Its diagonals cross at right angles. What is it?
Answer it in the portal, where your answer is checked as you type →
Symmetry in three dimensions Extended
A solid has a plane of symmetry if a flat plane cuts it into two halves that are mirror images of each other. It has an axis of rotational symmetry if it fits onto itself when turned less than a full turn about that line; the order is counted as for flat shapes.
Prisms. A right prism with a cross-section that has lines of symmetry has:
- planes of symmetry running along its length, one through each line of symmetry of the cross-section;
- 1 more plane across its middle, parallel to the two ends, because the two ends are identical.
So a cuboid with three different dimensions (cross-section a rectangle, 2 lines) has planes. A prism with an equilateral triangle cross-section has , and a regular hexagonal prism has . The order of rotational symmetry about the axis through the centres of the ends equals the order of the cross-section.
Square ends are a special case. A cuboid measuring 4 cm by 4 cm by 9 cm has a square cross-section, so planes. A cube has 9 planes: 3 parallel to the faces and 6 diagonal planes through pairs of opposite edges.
Pyramids. A right pyramid on a regular base with sides has planes of symmetry, all containing the vertical axis through the vertex, and rotational symmetry of order about that axis. There is no plane across the middle, because the vertex end and the base end are different.
Cylinders and cones. Every plane that contains the central axis is a plane of symmetry, so both have infinitely many planes of symmetry. A cylinder also has the plane across its middle; a cone does not. Each fits onto itself after any turn about its axis.
Worked example 3 Extended
Find the number of planes of symmetry of (a) a right prism whose cross-section is a regular pentagon, (b) a right pyramid with a regular hexagonal base. For (b), also give the order of rotational symmetry about the vertical axis.
(a) A regular pentagon has 5 lines of symmetry, giving 5 planes along the prism. Add the plane across the middle: 6 planes.
(b) A regular hexagon has 6 lines of symmetry, giving 6 planes through the vertex. A pyramid has no plane across its middle: 6 planes.
The hexagonal base fits onto itself every , so the pyramid has rotational symmetry of order 6 about its vertical axis.
Answer: (a) 6 (b) 6 planes, order 6
Check yourself Extended
Write down the number of planes of symmetry of a right prism whose cross-section is a square, given that the prism is not a cube.
Answer it in the portal, where your answer is checked as you type →
Check yourself Extended
A right pyramid has a regular base and rotational symmetry of order 5 about its vertical axis. How many planes of symmetry does it have?
Answer it in the portal, where your answer is checked as you type →
Key points
A line of symmetry splits a shape into two mirror-image halves; check with the folding test.
The order of rotational symmetry is the number of times a shape fits onto itself in one full turn. No rotational symmetry means order 1.
A regular polygon with sides has lines of symmetry and order ; its smallest angle of rotation is .
Square 4, 4; rectangle 2, 2; rhombus 2, 2; parallelogram 0, 2; kite 1, 1; isosceles trapezium 1, 1.
Symmetry swaps equal parts: base angles of an isosceles triangle, the equal pair of angles of a kite.
Extended: right prism: one plane per line of symmetry of the cross-section, plus one across the middle. Cube: 9.
Extended: right pyramid on a regular -sided base: planes, order . Cylinder and cone: infinitely many planes.
Where marks are lost
Counting the diagonals of a rectangle or a parallelogram as lines of symmetry.
Writing order 0 for a shape with no rotational symmetry. The answer is order 1.
Confusing the two counts: a parallelogram has no lines of symmetry but order 2.
Giving 2 lines of symmetry for a kite: only the diagonal through the vertices where equal sides meet is a line of symmetry.
On a grid, shading squares that give line symmetry when rotational symmetry was asked for, or the other way round.
Extended: adding a plane across the middle of a pyramid or cone. Only prisms and cylinders have one.
Extended: giving 3 planes for a cube. A cube also has 6 diagonal planes: 9 in all.
Related IGCSE Mathematics 0580 lessons
An original lesson written to the Cambridge IGCSE Mathematics 0580 syllabus (2025 to 2027). The worked examples and questions are our own, not Cambridge questions. Back to every IGCSE Mathematics 0580 lesson.