Rotational Symmetry (Cambridge (CIE) IGCSE Maths: Extended): Revision Note
Exam code: 0580 & 0980
Rotational symmetry
What is the order of rotational symmetry?
Rotational symmetry refers to the number of times a shape looks the same as it is rotated 360° about its centre
This number is called the order of rotational symmetry
Tracing paper can help work out the order of rotational symmetry
Draw an arrow on the tracing paper so you can easily tell when you have turned it through 360°
Notice that returning to the original shape contributes 1 to the order
This means a shape can never have order 0
A shape with rotational symmetry order 1 may be described as not having any rotational symmetry
The only time it looks the same is when you get back to the start
Examiner Tips and Tricks
Remember to use the trick above; using an upwards arrow on the tracing paper to show the starting orientation of the shape.
Worked Example
For the shape below, shade exactly 4 more squares so that the shape has rotational symmetry of order 4.
Answer:
The shape below appears the same 4 times if rotated through 360 degrees
Unlock more, it's free!
Join the 100,000+ Students that ❤️ Save My Exams
the (exam) results speak for themselves:
Was this revision note helpful?
Build on this topic
Questions students ask about Rotational Symmetry
Real questions from students revising this topic on Save My Exams
What is rotational symmetry?
Rotational symmetry is when a shape looks exactly the same as it started at some point during a full turn of 360° about its centre.
- Picture putting a pin through the centre of the shape and spinning it round once
- Every time it lands in a position that looks identical to the original, count 1
- The total count over one full turn is the order of rotational symmetry
- Returning to the start always counts as 1, so the lowest possible order is 1
Example from this page — the cross-shaped grid with shaded squares:
- Quarter turn: the shaded squares are in the wrong places, so it does not count
- Half turn: identical, so count 1
- Three-quarter turn: not quite the same, so it does not count
- Full turn: back to the start, so count 2
- The shape has rotational symmetry of order 2
Watch out: the outline can match while the shading does not. Every detail has to match the original before it counts.
How do you find the order of rotational symmetry of a shape?
Turn the shape through one full 360° about its centre and count how many times it looks exactly the same as the original — including the final return to the start.
Using tracing paper:
- Trace the shape and mark its centre
- Draw an upwards arrow on the tracing paper so you know when you have turned a full 360°
- Hold the centre still with your pen point and turn the tracing paper slowly
- Count each time the tracing fits exactly over the original
- Stop when the arrow points upwards again, and count that position too
Example from this page — the cross-shaped grid: after a quarter turn the shading does not match, after a half turn it does (1), after three-quarters it does not, and back at the start the arrow points up (2). The order is 2.
Common mistakes: forgetting to count the starting position, so your answer is one too small, and counting a position where the outline matches but the shading does not.
Can a shape have an order of rotational symmetry of 1?
Yes — and in fact every shape has an order of at least 1, because getting back to the start after a full turn always counts.
- A shape with order 1 only looks the same when it returns to its original position after 360°
- It never matches part-way round
- That is exactly what people mean by saying a shape has no rotational symmetry, so "order 1" and "no rotational symmetry" describe the same thing
What you cannot have is order 0, because that final return to the start always adds 1.
Most irregular shapes have order 1. The cross-shaped example on this page has order 2, because it matches at the half turn as well as at the full turn.
In the exam, if a shape only matches when the turn is complete, write 1, not 0. If your tracing-paper arrow is pointing up again and nothing matched along the way, the answer is 1.
