Symmetry
111. Spin & Match · Discovering rotational symmetry and order
The square fits onto itself exactly 4 times during a full rotation.
A figure has rotational symmetry of order n if it looks the same after a rotation of 360°/n about its centre. A square has order 4 (it fits onto itself at 90°, 180°, 270°, 360°).
What this lesson covers
Try to break it
Now drag R slowly between the green markers. The square tilts — it no longer fits its original outline! It only matches itself at the four green dots.
The proof, step by step
Prove that the square maps onto itself four times in one full turn.
- A figure has rotational symmetry if it coincides with itself during a rotation of less than 360°.
- The smallest angle of rotation that brings the figure back to its original position is the angle of rotational symmetry.
- The order of rotational symmetry is the number of times the figure fits onto itself during a full 360° rotation.
- For a square, this happens at 90°, 180°, 270°, and 360°, giving an order of 4.
Worked example
A regular hexagon is rotated about its centre. How many times does it fit onto itself during one complete rotation? What is its order of rotational symmetry?
A regular hexagon has 6 equal sides and angles. Rotating it by 60° brings it back to its original position. This happens 6 times in 360°. Hence, order is 6.
- 4 times, Order 4
- 5 times, Order 5
- 6 times, Order 6 — correct
- 3 times, Order 3