2D shapes
Rotational symmetry
A shape has rotational symmetry if it can be turned about a point to a new position and still look exactly the same. In this topic you learn to identify rotational symmetry and to work out its order.
What is rotational symmetry?
A shape has rotational symmetry if you can rotate it about a centre point to another position and it still looks the same as it did to start with. You can use tracing paper to help: trace the shape, then turn the tracing and see when it matches again.
The order of rotational symmetry
The order of rotational symmetry is the number of times a shape looks the same in one full turn. ā¢ A rectangle looks the same twice in a full turn, so it has order 2. ā¢ A button that matches itself four times has order 4. Order 1 means the shape only looks the same after a full turn ā it has no rotational symmetry beyond that. Every shape has an order of at least 1.
Rotational symmetry and lines of symmetry are different
A shape's order of rotational symmetry is not always the same as its number of lines of symmetry. For example, a parallelogram has 0 lines of symmetry but its order of rotational symmetry is 2. So a shape with no lines of symmetry can still have rotational symmetry.
Worked examples
Order of a rectangle
- Rotate a rectangle slowly about its centre point.
- It looks the same after a half turn, and again after a full turn.
- Count how many times it matched: twice.
Answer: A rectangle has rotational symmetry of order 2.
Order of a parallelogram
- Rotate the parallelogram about its centre point.
- It looks the same after a half turn and after a full turn.
- That is two matches in one full turn.
Answer: A parallelogram has rotational symmetry of order 2.
Order of an isosceles trapezium
- Rotate the isosceles trapezium about its centre point.
- It does not match itself until it comes right back to the start.
- That is only one match in a full turn.
Answer: An isosceles trapezium has rotational symmetry of order 1.
A higher order: the square
- Rotate a square about its centre point.
- It looks the same after a quarter turn, a half turn, a three-quarter turn and a full turn.
- Count the matches: four.
Answer: A square has rotational symmetry of order 4.
Lines of symmetry are not the order
- Marcus thinks every special quadrilateral has as many lines of symmetry as its order of rotational symmetry.
- A parallelogram has 0 lines of symmetry.
- But when you rotate it, it matches itself twice in a full turn ā order 2.
Answer: Marcus is wrong: a parallelogram has 0 lines of symmetry but order 2.
Where you see this in real life
- A roundabout in a park looks the same as it spins around.
- Many road signs and buttons look the same when you turn them.
- Turning a square tile a quarter turn leaves the pattern looking unchanged.
Key words
- Rotational symmetry
- When a shape can be turned about a point and still look the same.
- Order of rotational symmetry
- The number of times a shape looks the same in one full turn.
- Full turn
- Turning all the way around, back to the starting position.
- Line of symmetry
- A fold line where both halves of a shape match exactly (different from rotational symmetry).