Rotational Symmetry

What is Rotational Symmetry?

Some figures don’t have a line of symmetry (you can’t fold them), but they still look “balanced”. This usually means they have Rotational Symmetry.

If you rotate a figure around a fixed point (the Centre of Rotation), and it fits exactly onto itself before you complete a full turn ($360^{\circ}$), the figure has rotational symmetry.

Key Terms

1. Centre of Rotation: The fixed point about which the object turns.
2. Angle of Rotation (or Angle of Symmetry): The smallest angle you need to turn the shape for it to look the same.
3. Order of Symmetry: The number of times the shape looks like its original self in one full turn ($360^{\circ}$).

Formula

Examples

1. The Square

A square has rotational symmetry.

• Rotate $90^{\circ}$: Looks same.
• Rotate $180^{\circ}$: Looks same.
• Rotate $270^{\circ}$: Looks same.
• Rotate $360^{\circ}$: Back to start.
• Order: 4
• Angle: $90^{\circ}$

2. Equilateral Triangle

• Rotate $120^{\circ}$: Looks same.
• Rotate $240^{\circ}$: Looks same.
• Rotate $360^{\circ}$: Back to start.
• Order: 3
• Angle: $120^{\circ}$

3. Regular Hexagon

• Order: 6
• Angle: $360^{\circ} \div 6 = 60^{\circ}$

4. The Circle

A circle is special. You can rotate it by any angle, and it matches. It has an infinite order of rotational symmetry.

Figures with Both Reflection and Rotational Symmetry

Many regular shapes have both types of symmetry.

• Square: 4 lines of symmetry, Order 4 rotational symmetry.
• Circle: Infinite lines of symmetry, Infinite rotational symmetry.

However, some shapes like a Parallelogram (not a rhombus or rectangle) have rotational symmetry (Order 2) but no lines of symmetry.