Line & rotational symmetry
Theory
A line of symmetry folds a shape into two mirror halves, and a shape has rotational symmetry of order n if it matches itself n times in one full turn. A square has \(4\) lines of symmetry and order \(4\); the two ideas are separate.
A line of symmetry (also called an axis of symmetry) is a line that folds a shape into two halves that match exactly, like a mirror image.
A shape can have no lines, one line, or many lines of symmetry. A square has \(4\), a rectangle \(2\), an equilateral triangle \(3\), and a circle has infinitely many.
A shape has rotational symmetry of order \(n\) if it looks the same \(n\) times as it makes one full turn about its centre. A square is order \(4\), an equilateral triangle order \(3\), and a rectangle order \(2\).
Order \(1\) means the shape only matches itself after a whole turn, so it has no rotational symmetry. For any regular shape, both the number of lines and the order equal the number of sides.
Common shapes and their symmetry. For every regular polygon, the number of lines of symmetry and the order of rotational symmetry both equal the number of sides.
| Shape | Lines of symmetry | Rotational order |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Parallelogram | 0 | 2 |
| Rhombus | 2 | 2 |
| Kite | 1 | 1 |
| Regular pentagon | 5 | 5 |
| Regular hexagon | 6 | 6 |
| Circle | infinite | infinite |
How to find the symmetry of a shape
- Test each fold for line symmetry. Try the vertical, horizontal and diagonal folds. Count only the folds where one half is the mirror image of the other.
- Use the regular-shape rule. A regular polygon with \(n\) sides has exactly \(n\) lines of symmetry.
- Turn the shape for rotational symmetry. Rotate it about its centre through one full turn and count how many times it matches itself. That count is the order.
- Read order \(1\) as none. An order of \(1\) means the shape only fits after a full turn, so it has no rotational symmetry.
A regular shape has one axis for each side.
| Lines | \(=\) | \(6\) |
Three join opposite corners and three join the midpoints of opposite sides.
A half turn lands it back on itself; a quarter turn does not.
| Order | \(=\) | \(2\) |
No fold matches its two halves, but a half turn does.
| Lines | \(=\) | \(0\) |
| Order | \(=\) | \(2\) |
For a regular polygon both equal the number of sides.
| Lines | \(=\) | \(10\) |
| Order | \(=\) | \(10\) |
Common pitfalls
Frequently asked questions
What is a line of symmetry?
A line of symmetry is a line that folds a shape into two halves that are mirror images of each other. A square has four, a rectangle has two, and a circle has infinitely many.
What does the order of rotational symmetry mean?
The order counts how many times a shape looks exactly the same as it makes one full turn about its centre. A square has order \(4\) because a quarter, half, three-quarter and full turn all match.
Can a shape have rotational symmetry but no line of symmetry?
Yes. A parallelogram has no lines of symmetry, yet a half turn lands it back on itself, so its rotational order is \(2\). The letter S is the same. The two kinds of symmetry are separate.
How many lines of symmetry does a regular polygon have?
A regular polygon has one line of symmetry for each side. A regular pentagon has \(5\), a regular hexagon has \(6\), and the order of rotational symmetry matches the number of sides too.
Is a diagonal of a rectangle a line of symmetry?
No. The diagonal splits a rectangle into two equal triangles, but folding along it makes a long side land on a short side, so the halves do not match. A rectangle's two axes run through the midpoints of opposite sides.