Rotational symmetry
Theory
Rotational symmetry is a Year 4 maths shape skill: a shape has rotational symmetry if it looks exactly the same after a turn of less than a full turn about its centre. This page explains the order of rotational symmetry (how many times a shape matches in a full turn), the fraction of a turn to the first match, the quick rule for regular shapes, how to finish a half-turn pattern, and how rotational symmetry is different from line symmetry. In Western Australia this is taught in Year 5. Curriculum: VC2M4SP04 / AC9M4SP03.
A shape has rotational symmetry if it looks exactly the same after being turned less than a full turn about its centre. You turn the shape, rather than fold it.
The order of rotational symmetry is the number of times the shape looks the same in one full turn, counting the full turn at the end. A shape of order 4 first matches after a quarter of a turn, and a shape of order 2 first matches after a half turn.
Order 1 means no rotational symmetry. Every shape looks the same after a full turn, so an order of 1 does not count. This is different from line symmetry, which is about folding a shape into two matching halves. A shape can have one kind of symmetry without the other, or both.
These are the orders of rotational symmetry for the shapes you meet most often. They are worth remembering.
| Shape | Order of rotational symmetry |
|---|---|
| Square | 4 |
| Rectangle (not a square) | 2 |
| Rhombus | 2 |
| Slanted parallelogram | 2 |
| Equilateral triangle | 3 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Kite | 1 (none) |
| Isosceles triangle | 1 (none) |
| Scalene triangle | 1 (none) |
| Circle | more than you can count |
For any regular shape (all sides and corners the same), there is a quick rule:
The shape first looks the same after this fraction of a full turn:
How to find the order of rotational symmetry
- Find the centre. Mark the middle of the shape. This is the point you turn it about.
- Turn and count. Imagine turning the shape one full turn and count how many times it looks exactly the same, including the full turn at the end.
- Read off the order. That count is the order. If the only match is the full turn, the order is 1, which means no rotational symmetry. The first match happens after \(\dfrac{1}{\text{order}}\) of a turn.
How to finish a pattern with half-turn symmetry
- Find the centre. It is the point the pattern is turned about.
- Send each square across. For each shaded square, go through the centre and the same distance out the other side. Shade that square.
- Check. Turn the page upside down: the pattern should look exactly the same.
Each quarter turn moves every arm onto the place of the next arm, so the shape matches after \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(\dfrac{3}{4}\) and a full turn.
It matches \(4\) times, so its order is \(4\), and it first looks the same after \(\dfrac{1}{4}\) of a turn.
Turn N a half turn about its centre and it lands exactly on itself. It matches after a half turn and after a full turn, so its order is \(2\).
No fold makes its two halves match, so N has rotational symmetry but no line of symmetry.
A half turn sends each square to the opposite side of the centre, the same distance away.
The navy square in the top row, second from the left, goes to the bottom row, second from the right. The other navy square goes to the matching place below and to the left of the centre.
Each of the \(2\) navy squares needs \(1\) partner, so the finished pattern has \(2 + 2 = 4\) shaded squares.
It matches \(3\) times in a full turn, so its order is \(3\).
It first looks the same after \(\dfrac{1}{3}\) of a turn.
For a regular shape, the order is the same as the number of sides.
Its order is \(10\), and it first matches after \(\dfrac{1}{10}\) of a turn.
Common pitfalls
Frequently asked questions
What is rotational symmetry?
A shape has rotational symmetry if it looks exactly the same after being turned less than a full turn about its centre. You turn the shape rather than fold it.
What is the order of rotational symmetry?
The order of rotational symmetry is the number of times a shape looks the same during one full turn. A plus sign has order 4 because it matches after a quarter, half, three-quarter and full turn.
Can a shape have line symmetry but no rotational symmetry?
Yes. A heart shape has one line of symmetry down the middle, but it only looks the same after a full turn, so its order is 1 and it has no rotational symmetry.
What is the difference between rotational symmetry and line symmetry?
Rotational symmetry is about turning a shape about its centre so it looks the same. Line symmetry is about folding a shape so the two halves match. A shape can have one without the other, like the letter N, which has rotational symmetry but no line symmetry.
Why does order 1 mean no rotational symmetry?
Every shape looks the same after a whole turn, because it is back where it started. A shape only has rotational symmetry if it also matches somewhere before the full turn.
How do you finish a pattern so it looks the same after a half turn?
For each shaded square, go through the centre of the pattern and the same distance out the other side, and shade that square too.