Question
Determine the order of rotational symmetry of a square. Also state the angle of rotation at each step.
Solution — Step by Step
A shape has rotational symmetry if, when rotated about its centre, it looks exactly the same as the original before completing a full 360° rotation.
The order of rotational symmetry is the number of times the shape looks identical to itself during a complete 360° rotation (including the final position back to start).
Label the vertices of the square A, B, C, D (going clockwise from the top-left).
Now rotate the square about its centre:
- After 90°: The square looks the same (B is now where A was, C where B was, etc.)
- After 180°: Looks the same again
- After 270°: Looks the same again
- After 360°: Back to original position — looks the same
The square looks the same at 4 positions during a full 360° rotation:
- 90°
- 180°
- 270°
- 360° (= 0°, the original)
So the order of rotational symmetry = 4.
The angle of rotation =
The square has rotational symmetry at every 90° turn.
Why This Works
A square has 4-fold rotational symmetry because all 4 sides are equal and all 4 angles are equal (each 90°). Rotating by 90° maps each side and corner onto the next identical side and corner — the shape is indistinguishable from the original.
The general rule: if a regular polygon has sides, its order of rotational symmetry is , and the angle of rotation is .
| Shape | Sides | Order of symmetry | Angle of rotation |
|---|---|---|---|
| Equilateral triangle | 3 | 3 | 120° |
| Square | 4 | 4 | 90° |
| Regular pentagon | 5 | 5 | 72° |
| Regular hexagon | 6 | 6 | 60° |
| Circle | ∞ | ∞ | Any angle |
Alternative Method
You can also determine the order by asking: “How many times can I rotate this shape before it returns to the starting position, counting each identical appearance?”
For a square: rotate 1 quarter turn (looks same), rotate another (looks same), another (looks same), last quarter turn (back to start — looks same). That’s 4 times. Order = 4.
Common Mistake
Students sometimes confuse lines of symmetry with order of rotational symmetry. A square has 4 lines of symmetry (2 through midpoints of opposite sides, 2 through opposite corners) AND order of rotational symmetry 4. These numbers happen to be equal for a square, but they measure different things. A rectangle has 2 lines of symmetry but also has order of rotational symmetry 2 — the numbers still match here. However, an isosceles triangle has 1 line of symmetry but only order 1 rotational symmetry (it only looks the same at 360°). Don’t assume they’re always equal.