Line and rotational symmetry
Find mirror lines and rotational order by testing the whole shape.
Core and Extended.
Before you begin
- Recognise regular polygons.
- Divide 360° into equal turns.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many degrees make one full turn?
What you will learn
- Identify line symmetry of triangles, quadrilaterals and regular polygons.
- Find rotational order in a full turn.
- Distinguish a reflection from a rotation.
A mirror line matches both halves
A line of symmetry reflects every point of a shape onto another point of the same shape. Folding along that line makes the two halves coincide. A square has four lines: two through the midpoints of opposite sides and two diagonals. A non-square rectangle has two midpoint lines but neither diagonal is a symmetry line.
An equilateral triangle has three symmetry lines; a non-equilateral isosceles triangle has one through its apex and base midpoint; a scalene triangle has none. A non-square rhombus has two lines along its diagonals. A general parallelogram has none unless it has extra properties. Check the complete shape, including any pattern, labels or colouring specified in the question.
Count matches in one full turn
Rotational order counts how many times a shape coincides with itself during 360°, including the final full turn. A square has order four because 90°, 180°, 270° and 360° all match. A non-square rectangle has order two, matching at 180° and 360°. A shape with no smaller matching turn has order one, not zero.
A regular n-sided polygon has n symmetry lines and rotational order n. Its smallest positive matching turn is 360°/n: for a regular pentagon this is 72°. Line symmetry and rotational symmetry are different properties; a general parallelogram has order two even with no mirror line. A circle has infinitely many mirror lines and matches after any rotation, so it has no smallest positive matching angle.
Pause and explain
A non-square rectangle has which symmetry properties?
Worked example
State the line symmetries, rotational order and smallest matching turn of a regular hexagon.
Show the worked solution
- A regular six-sided polygon has six symmetry lines.
- It matches six times in one full turn, giving order six.
- Its smallest positive matching turn is 360 ÷ 6 = 60°.
Answer 6 lines; order 6; smallest turn 60°.
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.
Sketch a square and mark all its symmetry lines. State its rotational order.
Give me a hint
Test both diagonals as well as the midpoint lines.
Compare my reasoning
- Two lines join the midpoints of opposite sides.
- Both diagonals are also mirror lines, giving four total.
- Quarter-turns match the square, giving rotational order four.
4 symmetry lines; order 4.
Look for these in your work
- Included both diagonals.
- Counted full-turn matches rather than mirror lines alone.
For a regular decagon, find the number of mirror lines and smallest matching turn.
Give me a hint
Use n = 10 for a regular polygon.
Compare my reasoning
- There are ten symmetry lines.
- Rotational order is also ten.
- 360 ÷ 10 = 36°, the smallest positive matching rotation.
10 lines; 36° smallest turn.
Look for these in your work
- Used regularity, not just side count.
- Divided a full turn by the order.
A sign is a non-rectangular, non-rhombic parallelogram with no writing. Does it have a mirror line? Does it match after 180°?
Give me a hint
A reflection and a half-turn need separate tests.
Compare my reasoning
- The specified general parallelogram has no line of symmetry.
- A half-turn swaps each vertex with its opposite and preserves the shape.
- It therefore has zero mirror lines but rotational order two.
No mirror lines; matches after 180°; order 2.
Look for these in your work
- Did not confuse reflection with rotation.
- Used the restrictions excluding special parallelograms.
Common mistakes
- Counting symmetry of only part of a patterned shape.
- Giving a non-square rectangle diagonal mirror lines.
- Calling a shape's rotational order zero.
What can you explain now?
A shape only matches itself after a full 360° turn. What is its rotational order?
Compare with the explanation
1
The full turn still counts as one matching position; lack of a smaller turn does not give order zero.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, compare a square, a non-square rectangle and a general parallelogram. Test reflection and rotation separately.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
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