Line Symmetry
A shape has line symmetry (or reflective symmetry) if you can fold it so that both halves match exactly. The fold line is called the line of symmetry (or mirror line).
Triangle — 1 line
Rectangle — 2 lines
Circle — infinite lines
Lines of Symmetry — Common Shapes
| Shape | Lines of symmetry |
|---|---|
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Square | 4 |
| Rectangle | 2 |
| Regular hexagon | 6 |
| Circle | Infinite |
| Parallelogram | 0 |
Rotational Symmetry
A shape has rotational symmetry if it looks the same after being rotated less than a full turn. The order of rotational symmetry is how many times it fits its original outline in one full turn.
Square
A square looks the same after rotating 90°, 180°, 270°, and 360°.
Order of rotational symmetry = 4
Every shape has at least order 1 (full turn). We say a shape has rotational symmetry only if the order is 2 or more.
Real world: logo design
Many company logos exploit symmetry. The Mercedes-Benz three-pointed star has rotational symmetry of order 3 (looks the same after rotating 120°). The Adidas trefoil has rotational symmetry of order 3 and no line symmetry. Designers choose symmetry type deliberately to create a balanced, recognisable visual identity.
Real world: architecture and window design
Rose windows in Gothic cathedrals (e.g. Notre-Dame) are designed with rotational symmetry of order 8 or 12. Each petal is a reflection of its neighbour. Architects exploit high-order symmetry to create visually harmonious structures.
Real world: textile and fabric patterns
A tartan fabric pattern is built from a design that has line symmetry in both the horizontal and vertical directions. Fashion designers check for lines of symmetry to ensure the pattern aligns at seams when cut and sewn.
Practice Problems
How many lines of symmetry does a regular pentagon have?
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What is the order of rotational symmetry of an equilateral triangle?
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Does a parallelogram have line symmetry? Does it have rotational symmetry? Explain.