Two kinds of symmetry
Symmetry is about a shape matching itself. There are two ways this can happen, and a shape can have one, both, or neither.
- Line symmetry
- A shape has line symmetry (also called reflective symmetry) if you can draw a straight line so that one side is the exact mirror image of the other. That line is a line of symmetry.
- Rotational symmetry
- A shape has rotational symmetry if you can turn it about its centre by less than a full turn and it looks exactly the same as it did before.
Think of folding for the first kind and turning for the second. A plain rectangle, for example, folds neatly in two ways and also looks the same after a half turn, so it has both.
Lines of symmetry
To test a line of symmetry, imagine folding the shape along it. If the two halves land exactly on top of each other, it is a line of symmetry. A shape can have many, one, or none.
| Shape | Lines of symmetry |
|---|---|
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Rectangle (not a square) | 2 |
| Square | 4 |
| Parallelogram (general) | 0 |
A diagonal is not always a mirror line
It is tempting to draw a line of symmetry along the diagonal of a rectangle, but fold it and the halves do not match. A general parallelogram has no lines of symmetry at all, even though it looks tidy.
Rotational symmetry and order
Turn a shape a full 360 degrees about its centre and count how many times it looks exactly like the original (the final position back at the start counts once). That count is the order of rotational symmetry.
- Order of rotational symmetry
- The number of positions in one full turn where the shape looks unchanged. Every shape has at least order 1. If the order is 1, we say the shape has no rotational symmetry.
The angle you turn through between matching positions is:
A rectangle has rotational symmetry of order 2, so it matches itself every \(360 \div 2 = 180^\circ\).
Regular polygons
A regular polygon has all sides equal and all angles equal. This gives a lovely shortcut: a regular polygon with \(n\) sides has exactly \(n\) lines of symmetry and rotational symmetry of order \(n\).
State the number of lines of symmetry and the order of rotational symmetry of a regular pentagon, and the smallest angle it can be turned through to look the same.
Where this is assessed
Naming lines of symmetry and stating an order is Criterion A (Knowing and understanding). A clear labelled sketch with the mirror lines drawn earns Criterion C (Communicating). Deciding the symmetry of a real logo, tile or road sign is Criterion D (Applying maths in context).
Check yourself
1. How many lines of symmetry and what order of rotational symmetry does a square have? +
A square is a regular polygon with \(4\) sides, so it has 4 lines of symmetry and rotational symmetry of order 4 (matching every \(360 \div 4 = 90^\circ\)).
2. For a regular hexagon, state the lines of symmetry, the order of rotational symmetry, and the smallest turn that leaves it unchanged. +
A regular hexagon has \(n = 6\) sides, giving \(6\) lines of symmetry and order \(6\). The smallest turn is \(360 \div 6 = \mathbf{60^\circ}\), with 6 lines of symmetry and order 6.
3. How many lines of symmetry and what order of rotational symmetry does a general parallelogram (not a rectangle or rhombus) have? +
Folding never makes the halves match, so it has 0 lines of symmetry. A half turn about its centre does map it onto itself, so it has rotational symmetry of order 2.
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