What is symmetry?
A shape has symmetry when parts of it match exactly. Butterflies, snowflakes, faces and many logos are symmetrical. There are two main kinds: line symmetry and rotational symmetry.
Line (reflective) symmetry
A line of symmetry splits a shape into two halves that are mirror images. If you fold along the line, the halves match perfectly. If you place a mirror on the line, the reflection completes the shape.
| Shape | Lines of symmetry |
|---|---|
| Circle | infinitely many |
| Square | 4 |
| Rectangle (not square) | 2 |
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Regular hexagon | 6 |
| Parallelogram (not a rectangle) | 0 |
A regular polygon with n sides has n lines of symmetry.
Testing for line symmetry
- Trace and cut out the shape.
- Fold it along a possible line.
- If both halves match exactly, it’s a line of symmetry.
Or use a mirror on the line.
Symmetry in letters
| Letters | Symmetry |
|---|---|
| A, M, T, U, V, W, Y | one vertical line |
| B, C, D, E, K | one horizontal line |
| H, I, O, X | both vertical and horizontal |
| F, G, J, L, P, R | none |
(Depends on the font!)
Rotational symmetry
A shape has rotational symmetry if it looks the same after turning it less than a full turn. The order of rotational symmetry is how many times it matches itself in one full turn.
| Shape | Order |
|---|---|
| Square | 4 (every 90°) |
| Rectangle | 2 (every 180°) |
| Equilateral triangle | 3 (every 120°) |
| Regular hexagon | 6 |
| Letter S, letter Z | 2 |
A shape that only matches after a full 360° turn has order 1, which means no rotational symmetry.
Completing symmetrical drawings
On grid paper, reflect each point across the line of symmetry. A point 3 squares left of the line goes 3 squares right of it.
Worked examples
Example 1: How many lines of symmetry does a regular pentagon have?
5.
Example 2: Does the letter N have line symmetry?
No, but it has rotational symmetry of order 2.
Example 3: A shape matches itself every 60°. What is its order?
360 ÷ 60 = 6.
Common mistakes
- Thinking a rectangle’s diagonals are lines of symmetry. Fold one and the halves don’t match.
- Counting a full turn as rotational symmetry.
- Reflecting points the wrong distance from the line.
Try it yourself
- How many lines of symmetry does a rhombus (not a square) have?
- What is the order of rotational symmetry of a square?
- Does the letter H have line symmetry?
- Draw a shape with exactly one line of symmetry.
Answers: 1) 2 2) 4 3) yes, two lines 4) e.g. a heart or an isosceles triangle
Quick recap
- A line of symmetry makes two mirror image halves.
- Rotational symmetry means the shape matches itself during a turn.
- The order of rotational symmetry counts matches in one full turn.