What is rotational symmetry?
A shape has rotational symmetry if it looks exactly the same after being turned through less than a full turn (360°) about a fixed point called the centre of rotation. The number of times the shape fits exactly onto itself during one complete turn is called the order of rotational symmetry.
Every shape fits onto itself once after a full 360° turn, so the smallest possible order is 1 (this means it has no rotational symmetry beyond the trivial full turn).
Key idea / 要点
Angle of rotation for each fit = 360° ÷ order. For example, order 4 means the shape fits onto itself every 360° ÷ 4 = 90°.
Finding the order
Turn the shape slowly through a full turn and count how many times it matches its original outline.
Worked example / 例题
Find the order of rotational symmetry of a regular hexagon, and the smallest angle of rotation.
A regular hexagon has 6 equal sides, so it fits onto itself 6 times in one full turn. Order = 6.
Smallest angle = 360° ÷ 6 = 60°.
A regular polygon with n sides has rotational symmetry of order n. An equilateral triangle has order 3; a square has order 4.
Remember / 记住
- Order = number of fits in one full turn.
- Angle of rotation = 360° ÷ order.
- A regular n-sided polygon has order n.
- Order 1 means no true rotational symmetry.