What is a regular polygon?
A polygon is a closed shape made of straight sides. A regular polygon is special: all its sides are the same length and all its interior angles are equal. Common examples are the equilateral triangle (3 sides), the square (4 sides) and the regular hexagon (6 sides).
An irregular polygon has sides or angles that are not all equal, such as a rectangle (equal angles but unequal sides) or a general triangle.
Key idea / Key idea
A polygon with n sides has n lines of symmetry only if it is regular, and it has rotational symmetry of order n.
Symmetry of regular polygons
Every regular polygon with n sides has exactly n axes (lines) of symmetry and rotational symmetry of order n. For example, a regular pentagon (5 sides) has 5 lines of symmetry and rotational symmetry of order 5.
| Polygon | Sides | Lines of symmetry |
| Equilateral triangle | 3 | 3 |
| Square | 4 | 4 |
| Regular hexagon | 6 | 6 |
Constructing regular polygons
Because all sides and angles are equal, a regular polygon can be drawn by dividing a circle into equal parts. Dividing 360° by n gives the angle at the centre for each side.
Worked example / Worked example
Find the angle at the centre between two adjacent vertices of a regular octagon (8 sides).
Central angle = 360° ÷ 8 = 45°. So the 8 vertices are spaced 45° apart around the centre.
Remember / Remember
- Regular = all sides equal AND all angles equal.
- n sides → n lines of symmetry.
- Rotational symmetry order = number of sides.
- Central angle = 360° ÷ n.