Recognise line and rotational symmetry
Identify reflection axes and rotational order in plane figures, then extend symmetry to solids.
Section 5 · Geometry
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a line of symmetry do?
What you will learn
- Recognise line symmetry in two-dimensional shapes
- Find rotational symmetry order
- Recognise symmetry planes and axes in solids
A symmetry line reflects matching halves
A reflection axis divides a shape into mirror images. It may pass through vertices, sides or both; test the full outline.
A regular n-sided polygon has n lines of symmetry. Other shapes may have none, one or several.
Rotational order counts matching turns
Order is the number of times a shape maps onto itself during one full 360° turn. A square has order 4; an equilateral triangle has order 3.
Extended candidates also identify symmetry planes or axes in prisms, cylinders, pyramids and cones. A plane divides reflected halves; an axis is a line about which rotation maps the solid onto itself.
Pause and explain
How is rotational order counted?
Worked example
State the line-symmetry count and rotational order of a square.
Show the worked solution
- It has vertical, horizontal and two diagonal reflection axes.
- It maps onto itself every 90° during a full turn.
- There are four matching orientations.
Answer 4 lines of symmetry; rotational order 4.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A regular hexagon is rotated through 360°. Find its rotational order and smallest matching turn.
Give me a hint
A regular n-gon matches n times; divide 360° by n.
Compare my reasoning
- It has six matching orientations.
- Order = 6.
- Smallest matching turn = 360°/6 = 60°.
Order 6; turn 60°.
Look for these in your work
- Orientations counted
- Full turn used
- Angle calculated
A non-square rectangle has how many reflection axes and what rotational order?
Give me a hint
Check its two midlines and the smallest turn that returns it to itself.
Compare my reasoning
- It has one vertical and one horizontal reflection axis.
- A half-turn maps it onto itself.
- Its rotational order is 2.
2 lines; rotational order 2.
Look for these in your work
- Axes identified
- Quarter-turn rejected
- Half-turn counted
A right circular cylinder has matching circular ends. Describe a plane and an axis of symmetry, and explain how a fixed handle could change them.
Give me a hint
Consider a plane halfway through its height and rotations around its central axis.
Compare my reasoning
- A horizontal plane halfway between the ends reflects the cylinder onto itself.
- Rotation about its central axis maps it onto itself.
- A fixed handle may break these symmetries, so attachments must be included in the model.
Mid-height plane and central rotational axis; a handle may break symmetry.
Look for these in your work
- Plane located
- Axis described
- Model limitation included
Common mistakes
- Counting the starting position as an extra turn
- Confusing order with turn angle
- Drawing a line that matches only part of the outline
- Calling a solid's plane of symmetry an edge
What can you explain now?
What is the rotational order of an equilateral triangle?
Compare with the explanation
3
It maps onto itself three times in one full turn.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do reflection axes, symmetry planes and rotational order describe different symmetries?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.