Symmetry – Balance and Mirror Images in Shapes
Symmetry is the quality of balance and regularity. A shape has symmetry if it looks the same after a transformation — a fold, a turn, or a slide. Symmetry appears in nature, art, architecture, and mathematics.
Line Symmetry (Reflective Symmetry)
A shape has line symmetry (also called reflective symmetry or a line of symmetry) if you can fold it along a straight line so both halves match exactly. That fold line is called the axis of symmetry.
Lines of Symmetry in Common Shapes
| Shape | Lines of Symmetry |
|---|---|
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Square | 4 |
| Rectangle | 2 |
| Rhombus | 2 |
| Parallelogram | 0 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinitely many |
Rotational Symmetry
A shape has rotational symmetry if it can be rotated about a central point by less than 360° and still look exactly the same. The number of times it fits in one full turn is called the order of rotational symmetry.
Order of Rotational Symmetry
| Shape | Order | Angle of Rotation |
|---|---|---|
| Equilateral triangle | 3 | 120° |
| Square | 4 | 90° |
| Rectangle | 2 | 180° |
| Regular hexagon | 6 | 60° |
| Scalene triangle | 1 (none) | 360° only |
| Circle | Infinite | Any angle |
Point Symmetry
A shape has point symmetry if every part has a matching part at an equal distance through the centre. This is the same as rotational symmetry of order 2 (the shape looks the same after a 180° rotation). Parallelograms and rectangles have point symmetry.
Real-Life Examples
- A butterfly has one line of symmetry (left-right).
- A snowflake has six lines of symmetry and rotational symmetry of order 6.
- The letter H has two lines of symmetry; the letter S has point symmetry.
- A starfish typically has five lines of symmetry.
Key Takeaways
- Line symmetry: a shape maps onto itself when folded about a line.
- A regular n-sided polygon has n lines of symmetry.
- Rotational symmetry order = number of times the shape maps onto itself in one full rotation.
- Every shape has rotational symmetry of at least order 1 (itself at 360°).
Practice Questions
- How many lines of symmetry does a regular octagon have?
- State the order of rotational symmetry of a regular pentagon.
- Does a parallelogram have any lines of symmetry? Does it have rotational symmetry?
- Draw a shape with exactly two lines of symmetry and rotational symmetry of order 2.
- Which capital letters of the alphabet have exactly one line of symmetry? List at least five.