Quick revision notes · Mathematics 4024 · Chapter 18 of 35
Symmetry
Lines of symmetry, rotational symmetry, symmetric quadrilaterals and planes of symmetry in solids.
4 sectionsWorked examples
Common mistakes
At a glance
Lines of symmetry, rotational symmetry, symmetric quadrilaterals and planes of symmetry in solids.
1Line symmetry
- A line of symmetry splits a shape into two mirror images.
- Test by folding (or with a mirror or tracing paper).
- A regular polygon with n sides has n lines of symmetry.
Worked example
How many lines of symmetry does a regular hexagon have?
- Regular n-gon: n lines.
6
2Rotational symmetry
- The order is how many times a shape looks the same in one full turn.
- Order 1 means no rotational symmetry (it only matches itself after 360°).
- A regular n-gon has rotational symmetry of order n.
Worked example
What is the order of rotational symmetry of a parallelogram?
- It looks the same after a half turn (180°) and a full turn.
Order 2
3Symmetry of triangles and quadrilaterals
- Know these well — they are used to find missing angles and lengths.
| Shape | Lines | Order |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 (the diagonals) | 2 |
| Parallelogram | 0 | 2 |
| Kite | 1 | 1 |
| Isosceles trapezium | 1 | 1 |
Worked example
Which quadrilateral has rotational symmetry of order 2 but no lines of symmetry?
- Rectangle and rhombus have lines; parallelogram has none.
Parallelogram
4Symmetry in solids
- A plane of symmetry cuts a solid into two mirror halves.
- Cuboid (all edges different): 3 planes. Cube: 9 planes.
- Prisms, cylinders, pyramids and cones: picture slicing through the middle.
Worked example
How many planes of symmetry does a cuboid with three different edge lengths have?
- One plane through the middle of each pair of opposite faces.
3
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| A parallelogram has 2 lines of symmetry | It has none; its order of rotation is 2 |
| Order 0 rotational symmetry | The smallest order is 1 |
| A rhombus's lines go through the midpoints of its sides | Its lines are the diagonals |
★In the exam
- Use tracing paper in the exam to check lines and order.
- Complete-the-pattern questions: reflect each square across the given line(s).
- Answer with a number for 'order' — 'order 2', not 'twice'.