Syllabus coverage
- Recognise line symmetry in two dimensions
- Recognise the order of rotational symmetry of a two-dimensional shape
- Use the properties of triangles, quadrilaterals and polygons that are directly related to their symmetries
- Extended only: recognise the symmetry properties of prisms, cylinders, pyramids and cones, including their planes and axes of symmetry
Part of the IGCSE Maths syllabus 0580.
Line symmetry: the fold line
A shape has line symmetry if you can fold it so that the two halves land exactly on each other. The fold is the line of symmetry, also called the mirror line or the axis of symmetry. Line symmetry is the same thing as reflection symmetry: each half is the mirror image of the other.
A line of symmetry always passes through the centre of the shape and cuts it into two parts of exactly the same size and shape. Every vertex must have a partner the same distance from the line on the other side; a vertex that lies on the line stays where it is.
A shape can have no lines of symmetry, one line, or many. A scalene triangle has none, an isosceles triangle has one, a square has four, and a circle has infinitely many, because every line through the centre of a circle folds it in half exactly.
Finding ALL the lines of symmetry
Questions ask for the number of lines of symmetry, or for the lines to be drawn on the diagram. Work through the candidates in the same order every time so that none is missed: the vertical line through the centre, the horizontal line through the centre, the two diagonals, and then any other line you suspect.
Two checks keep you honest. A line of symmetry must map the shape onto itself, so it either passes through a vertex and the opposite vertex, or through the midpoint of a side and the midpoint of the opposite side - if a candidate line does neither, it cannot be a line of symmetry. And a line of symmetry must pass through the centre, so any line that misses the centre is wrong immediately.
Two lines of symmetry at right angles force more symmetry than you might expect: reflecting in one and then in the other is the same as a half turn, so the shape also has rotational symmetry of order 2. That is why a rectangle, with 2 lines of symmetry, always has order 2.
| Shape | Lines of symmetry |
|---|---|
| Scalene triangle | 0 |
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Rectangle that is not a square | 2 |
| Square | 4 |
| Kite | 1 |
| Rhombus | 2 |
| Parallelogram | 0 |
| Isosceles trapezium | 1 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | infinitely many |
Rotational symmetry and its order
A shape has rotational symmetry if it looks exactly the same at some point during a turn about its centre. The order is the number of positions in a full 360° turn in which the shape looks the same, counting the starting position. A shape that only looks right where it started has order 1, which means it has no rotational symmetry.
The order is a whole number, and the turn that works is 360° divided by the order: order 2 means a half turn, order 3 a third of a turn (120°), order 4 a quarter turn (90°) and order 6 a sixth of a turn (60°). The centre of rotation is the point you turn the shape about, and for a shape with two lines of symmetry it is the point where they cross.
The safe method is tracing paper. Trace the shape, put your pencil point on the centre, and turn the tracing until the shape lines up again. Count every position where it lines up, including the first one, and the count is the order.
| Turn that leaves the shape looking the same | Order of rotational symmetry |
|---|---|
| a full turn of 360° only | 1 |
| half turn, 180° | 2 |
| third of a turn, 120° | 3 |
| quarter turn, 90° | 4 |
| sixth of a turn, 60° | 6 |
The symmetry of triangles and quadrilaterals
The symmetry properties of the special quadrilaterals are the ones the papers test, and they separate the shapes neatly. A square has 4 lines of symmetry and order 4. A rectangle and a rhombus each have 2 lines of symmetry and order 2, but the lines are different: the rectangle's two lines join the midpoints of opposite sides, while the rhombus's two lines are its diagonals. A parallelogram has no lines of symmetry at all, yet it does have rotational symmetry of order 2. A kite and an isosceles trapezium each have 1 line of symmetry and order 1.
Among triangles, the equilateral triangle has 3 lines and order 3, the isosceles triangle has 1 line and order 1, and the scalene triangle has none and order 1. A right-angled triangle has no line of symmetry unless it is also isosceles, in which case it has exactly 1 - the line through the right angle and the midpoint of the hypotenuse.
Notice that the two properties do not come in a pair. Rotational symmetry does not give you line symmetry, and one line of symmetry does not give you any rotational symmetry beyond order 1. Answer each part of a symmetry question separately.
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Scalene triangle | 0 | 1 |
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Parallelogram | 0 | 2 |
| Kite | 1 | 1 |
| Isosceles trapezium | 1 | 1 |
| Trapezium that is not isosceles | 0 | 1 |
Regular polygons, and shapes that are not regular
A regular polygon has every side equal and every interior angle equal, and its symmetry follows straight from that: a regular polygon with sides has exactly lines of symmetry and rotational symmetry of order . A regular pentagon has 5 lines and order 5, a regular hexagon 6 and 6, a regular octagon 8 and 8, and a regular decagon 10 and 10.
Each line of symmetry of a regular polygon passes either through a vertex and the midpoint of the opposite side, or through two opposite vertices. In a regular hexagon three of the six lines pass through opposite vertices and three through the midpoints of opposite sides, which is a good check that you have found all six.
Irregular shapes have fewer lines, and you have to look at the shape rather than count the sides. An irregular hexagon might have 1 line of symmetry or none at all. If a question gives you an irregular polygon, test each candidate line by folding it mentally or by tracing the shape onto paper.
Letters, patterns, and symmetry in three dimensions
Capital letters are used as ready-made shapes. A letter's symmetry depends on how it is printed, so answer from the letter as drawn and assume the plain block capitals used in the paper. An A has 1 line of symmetry (vertical) and order 1. A B has 1 line (horizontal) and order 1. An H has 2 lines, vertical and horizontal, and order 2. A T has 1 line (vertical) and order 1. An N and an S have no lines of symmetry but do look the same after a half turn, so each has order 2. Letters such as F, G, J, L, P, Q and R have neither.
A pattern behaves like any other shape. A motif repeated around a centre has rotational symmetry of order equal to the number of repeats, and usually a line of symmetry through each repeat. A pattern made by reflecting a motif in a line has a line of symmetry along that line.
Extended candidates also need symmetry in three dimensions. A plane of symmetry cuts a solid into two halves that are mirror images of each other. A cube has 9 planes of symmetry: 3 parallel to pairs of faces and 6 passing through pairs of opposite edges. A cuboid whose three edge lengths are all different has 3 planes. A cylinder has infinitely many planes of symmetry, each containing its axis, and a cone has infinitely many too, each containing its axis. A square-based pyramid has 4 planes of symmetry, each passing through the apex, and an axis of symmetry running from the apex to the centre of the base. A cylinder and a cone each have one axis of symmetry, running through the centre from one end to the other.
| Block capital letter | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| A | 1 (vertical) | 1 |
| B | 1 (horizontal) | 1 |
| H | 2 (vertical and horizontal) | 2 |
| N | 0 | 2 |
| O | 2 (vertical and horizontal) | 2 |
| S | 0 | 2 |
| T | 1 (vertical) | 1 |
| F, G, J, L, P, Q, R | 0 | 1 |
Key terms you must be able to define
- Line symmetry
- A shape has line symmetry if it can be folded so that the two halves land exactly on each other.
- Line of symmetry
- The fold line that cuts a shape into two matching halves. Also called the mirror line or the axis of symmetry.
- Mirror line
- Another name for a line of symmetry: each half of the shape is the mirror image of the other.
- Reflection
- The mirror image of a shape in a line. Line symmetry means the shape is its own reflection.
- Rotational symmetry
- A shape has rotational symmetry if it looks exactly the same at some point during a turn about its centre.
- Order of rotational symmetry
- The number of positions in a full turn in which the shape looks the same, counting the starting position. Order 1 means no rotational symmetry.
- Centre of rotation
- The point a shape is turned about. For a shape with two lines of symmetry it is where the lines cross.
- Regular polygon
- A polygon with every side equal and every interior angle equal. A regular polygon with n sides has n lines of symmetry and rotational order n.
- Irregular polygon
- A polygon that is not regular, so its symmetry has to be found from the shape itself.
- Square
- 4 lines of symmetry - vertical, horizontal and both diagonals - and rotational symmetry of order 4.
- Rectangle
- 2 lines of symmetry, joining the midpoints of opposite sides, and rotational symmetry of order 2. Its diagonals are not lines of symmetry.
- Rhombus
- 2 lines of symmetry, which are its diagonals, and rotational symmetry of order 2.
- Parallelogram
- No lines of symmetry, but rotational symmetry of order 2.
- Kite
- 1 line of symmetry, the diagonal joining the two vertices where the equal sides meet, and rotational symmetry of order 1.
- Isosceles trapezium
- A trapezium with the two non-parallel sides equal; it has 1 line of symmetry and rotational symmetry of order 1.
- Equilateral triangle
- 3 lines of symmetry and rotational symmetry of order 3.
- Isosceles triangle
- 1 line of symmetry and rotational symmetry of order 1.
- Scalene triangle
- No lines of symmetry and rotational symmetry of order 1.
- Plane of symmetry
- A plane that cuts a solid into two halves that are mirror images of each other. Extended only.
- Axis of symmetry
- A line about which a solid can be turned and still look the same. Extended only.
In context
Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.
A rectangle that is not a square has 2 lines of symmetry: the vertical line joining the midpoints of the top and bottom sides, and the horizontal line joining the midpoints of the left and right sides. Its diagonals are not lines of symmetry, because folding along a diagonal does not put the rectangle onto itself. Its order of rotational symmetry is 2, since a half turn maps it onto itself.
A parallelogram is the standard example. A half turn about the point where the diagonals cross maps each vertex onto the opposite vertex, so the order is 2. But no fold makes the two halves match: the diagonals do not work and neither do the lines joining the midpoints of opposite sides, so there are 0 lines of symmetry.
A regular hexagon has 6 lines of symmetry and rotational symmetry of order 6. Three of the lines pass through opposite vertices and three through the midpoints of opposite sides, and each of them folds the hexagon into two matching halves. A sixth of a turn (60°) brings it back onto itself.
The block capital H has a vertical line of symmetry and a horizontal one, so 2 lines, and it looks the same after a half turn, so its order is 2. The letter N has no lines of symmetry, because no fold matches its two halves, but a half turn does map it onto itself, so its order is 2 as well.
Flashcards
Click a card to flip it. Mark honestly — the tracker uses this.
Exam-style questions
6 questions, 13 marks. Write your answer, then reveal the mark scheme and tick each point you actually made. Answering honestly here is what makes the rest of the site useful.
Mark scheme — tick each point you actually made
- B12, because a half turn about the centre maps the parallelogram onto itselfnot checked yet
Mark scheme — tick each point you actually made
- B1
- B11 line of symmetry for the kite, along the diagonal joining the two vertices where the equal sides meetnot checked yet
Mark scheme — tick each point you actually made
- B1
- B1parallelogramnot checked yet
Mark scheme — tick each point you actually made
- B1
- B10 lines of symmetry, because no fold makes the two halves matchnot checked yet
Mark scheme — tick each point you actually made
- M1
- A112 sidesnot checked yet
Mark scheme — tick each point you actually made
- B1
- B1
- B1
- B1
Exam technique for this topic
- Draw the lines of symmetry on the diagram, one line per mark. A question that says 'draw all the lines of symmetry' gives a mark for each correct line, so a bare '4' in the margin scores nothing.
- Test the diagonals as well as the vertical and horizontal lines. The commonest lost mark in this topic is a square answered with 2 lines instead of 4, and a rhombus answered with 2 lines that are the wrong two - a rhombus's lines of symmetry are its diagonals.
- Remember that an order of 1 means no rotational symmetry, not zero. A scalene triangle, a kite and the letter T all have order 1, and writing 0 for the order is a wrong answer.
- Line symmetry and rotational symmetry are independent, so answer the two parts separately. A parallelogram has 0 lines and order 2, while an isosceles triangle has 1 line and order 1.