Chapter 7 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

M3.5.7 Chapter 7

Turning symmetry

A square can be turned and still look the same. How many times does it fit on itself in one full turn?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the order of turning symmetry of a shape
  • Make a pattern with turning symmetry of order 4

A shape has turning symmetry when it fits on itself after a part turn. You turn it about a point in its middle. That point is the centre of turning.

Draw it first

Turn a square a quarter turn

  1. A square with a dot at its centre. Step 2: the corner A at the top right is marked, and the square is turned a quarter turn, 90°, so A moves to A' at the top left. A curved arrow shows the turn. Step 3: the turned square lies exactly on the first. The labels are A, A' and 90°.

    1 Draw the square and mark its centre with a dot.

  2. A square with a dot at its centre. Step 2: the corner A at the top right is marked, and the square is turned a quarter turn, 90°, so A moves to A' at the top left. A curved arrow shows the turn. Step 3: the turned square lies exactly on the first. The labels are A, A' and 90°.AA'90°

    2 Mark one corner A. Turn the square a quarter turn, 90°, about the centre. Corner A moves to A'. The turned square fits exactly on the first.

The order of turning symmetry is how many times a shape fits on itself in one full turn. A square fits 4 times, so its order is 4. The smallest turn that fits is 360° ÷ 4 = 90°.

A square with a dot at its centre. Four numbered dots, 1, 2, 3 and 4, mark where one corner lands after each quarter turn. The order is 4.1234
The marked corner lands on a corner 4 times, so the order is 4.

Worked example

What is the order of turning symmetry of a rectangle that is not a square?

  1. Mark one corner of the rectangle.
  2. Turn the rectangle a half turn, 180°. It fits on itself.
  3. A quarter turn does not fit. The long sides would land on the short sides.
  4. After the full turn it fits again.
  5. It fits 2 times in one full turn, so the order is 2.

Answer: 2

A rectangle with a dot at its centre. Two numbered dots, 1 and 2, mark where one corner lands after each half turn. The order is 2.12
The marked corner lands on a corner 2 times, so the order is 2.

Worked example

A regular hexagon fits on itself in a full turn. What is its order, and the smallest turn?

  1. A regular hexagon has 6 equal sides and 6 equal angles.
  2. Each corner can move to the next corner and the shape still fits.
  3. So it fits 6 times in one full turn.
  4. The order is 6.
  5. The smallest turn is 360° ÷ 6 = 60°.

Answer: Order 6 and a smallest turn of 60°

A regular hexagon with a dot at its centre. Six numbered dots, 1 to 6, mark where one corner lands after each turn of 60°. The order is 6.123456
Six turns of 60° make one full turn of 360°.

Some shapes fit only after the full turn. Their order is 1. We say they have no turning symmetry. An isosceles triangle is one. A shape can have turning symmetry and no line of symmetry.

A table with three columns: shape, lines and order. The square has 4 lines and order 4. The rectangle has 2 lines and order 2. The parallelogram has 0 lines and order 2. The kite has 1 line and order 1. The letter Z has 0 lines and order 2.shapelinesordersquare44rectangle22parallelogram02kite11letter Z02
Lines of symmetry and the order of turning symmetry are different counts.

You can use turning symmetry to make a pattern. Draw one shape. Copy it round a centre point, one turn at a time. For order 4, each copy is turned 90° from the last. You draw 4 copies in all.

Draw it first

Make a pattern of order 4

  1. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    1 Mark a centre point. Draw a shape beside it.

  2. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    2 Turn the paper 90° about the centre. Draw the shape again.

  3. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    3 Turn another 90°. Draw the shape a third time.

  4. A pattern built round a centre dot. Step 1: one triangle to the right of the centre. Step 2: a second triangle turned 90°. Step 3: a third triangle turned 180°. Step 4: a fourth triangle turned 270°. The finished pattern has order 4.

    4 Turn another 90°. Draw it a fourth time. The pattern has order 4.

See it three ways

With things

Trace a square on paper. Hold the tracing on the square with a pencil point at its centre. Turn the tracing a quarter turn.

As a picture

A square with a dot at its centre. The corner A moves to A' after a turn of 90°, shown by a curved arrow. The turned square lies on the first. The labels are A, A' and 90°.AA'90°

In symbols

Order 4. The smallest turn is 360° ÷ 4 = 90°.

Try it

Find the order with tracing paper

You need: a shape drawn on paper, such as a triangle with equal sides, a sheet of thin paper, a pencil.

  1. Lay the thin paper on the shape. Trace the shape and mark its centre.
  2. Hold the tracing on the shape with a pencil point on the centre.
  3. Turn the tracing slowly. Each time it fits on the shape, count 1.
  4. Stop when the tracing is back where it started. The count is the order.

What you will see: A triangle with equal sides fits 3 times in a full turn, so its order is 3. A rectangle fits 2 times.

If you cannot try it, this is what happens.
A triangle with equal sides and a dot at its centre. Three numbered dots, 1, 2 and 3, mark where one corner lands after each turn of 120°. The order is 3.123

A wrong idea many people have

A parallelogram has no symmetry.

Why it feels right: It has no line of symmetry, so it seems to have no symmetry at all.

The test that breaks it: Trace a parallelogram. Hold the tracing at the centre and turn it a half turn. It fits on the shape.

What is true: A parallelogram has turning symmetry of order 2. Line symmetry and turning symmetry are different.

Key words

turning symmetry
a shape has it when it fits on itself after a part turn.
centre of turning
the point a shape is turned about.
order
how many times a shape fits on itself in one full turn.

Back to your guess

The answer is A: Four times, every quarter turn.

Turn a square about its centre. After a quarter turn it fits on itself. It fits again after a half turn, after three quarters of a turn, and after the full turn. That is 4 times.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the order of turning symmetry of a triangle with three equal sides?

    A triangle with three equal sides and a dot at its centre.
    Tap your answer.

    1. Not this one. Order 1 means it fits only after the full turn. This triangle also fits after a turn of 120°. Go to M3.5.7. Try another.

    2. Not this one. There are 3 corners, and each turn moves a corner to the next corner. So there are 3 turns, not 6. Go to M3.5.7. Try another.

    3. Yes, that is right. The triangle fits on itself after 120°, 240° and 360°. That is 3 times, so the order is 3.

    See the answer

    3

    The triangle fits on itself after 120°, 240° and 360°. That is 3 times, so the order is 3.

    • If you chose A: Order 1 means it fits only after the full turn. This triangle also fits after a turn of 120°. Go to M3.5.7.
    • If you chose B: There are 3 corners, and each turn moves a corner to the next corner. So there are 3 turns, not 6. Go to M3.5.7.

  2. Which shape has turning symmetry of order 2 but no line of symmetry?

    Three shapes with their names: a rectangle and an isosceles triangle on top, and a parallelogram below them.rectangleisosceles triangleparallelogram
    Tap your answer.

    1. Not this one. A rectangle has order 2, but it also has 2 lines of symmetry. Go to M3.5.6. Try another.

    2. Yes, that is right. A parallelogram fits on itself after a half turn, so its order is 2. No fold makes its halves match, so it has no line of symmetry.

    3. Not this one. An isosceles triangle has 1 line of symmetry and its order is 1. Go to M3.5.7. Try another.

    See the answer

    Parallelogram

    A parallelogram fits on itself after a half turn, so its order is 2. No fold makes its halves match, so it has no line of symmetry.

    • If you chose A: A rectangle has order 2, but it also has 2 lines of symmetry. Go to M3.5.6.
    • If you chose C: An isosceles triangle has 1 line of symmetry and its order is 1. Go to M3.5.7.

  3. A shape fits on itself 5 times in one full turn. What is the smallest turn that fits it?

    A regular pentagon with a dot at its centre. Five numbered dots, 1 to 5, mark where one corner lands after each turn.12345
    Tap your answer.

    1. Yes, that is right. The 5 turns make one full turn of 360°. Each turn is 360° ÷ 5 = 72°.

    2. Not this one. That is the turn for order 4. Divide 360° by 5. Go to M3.5.7. Try another.

    3. Not this one. That is the turn for order 6. Divide 360° by 5. Go to M3.5.7. Try another.

    See the answer

    72°

    The 5 turns make one full turn of 360°. Each turn is 360° ÷ 5 = 72°.

    • If you chose B: That is the turn for order 4. Divide 360° by 5. Go to M3.5.7.
    • If you chose C: That is the turn for order 6. Divide 360° by 5. Go to M3.5.7.

  4. You copy a shape 4 times round a centre to make a pattern of order 4. How far do you turn it each time?

    A pattern with a dot at its centre and two triangles, the second turned from the first. The pattern is not finished.
    Tap your answer.

    1. Not this one. That would give 8 copies in a full turn. For 4 copies divide 360° by 4. Go to M3.5.7. Try another.

    2. Not this one. That is a half turn. It would give only 2 copies in a full turn. Go to M3.5.7. Try another.

    3. Yes, that is right. Four copies make one full turn of 360°. Each turn is 360° ÷ 4 = 90°.

    See the answer

    90°

    Four copies make one full turn of 360°. Each turn is 360° ÷ 4 = 90°.

    • If you chose A: That would give 8 copies in a full turn. For 4 copies divide 360° by 4. Go to M3.5.7.
    • If you chose B: That is a half turn. It would give only 2 copies in a full turn. Go to M3.5.7.

  5. Which shape has a line of symmetry but no turning symmetry?

    Three shapes with their names: a square and an isosceles triangle on top, and a parallelogram below them.squareisosceles triangleparallelogram
    Tap your answer.

    1. Not this one. A square has 4 lines of symmetry and also turning symmetry of order 4. Go to M3.5.7. Try another.

    2. Yes, that is right. An isosceles triangle has 1 line of symmetry. It fits on itself only after the full turn, so its order is 1.

    3. Not this one. A parallelogram has turning symmetry of order 2 but no line of symmetry. This question asks the other way round. Go to M3.5.7. Try another.

    See the answer

    Isosceles triangle

    An isosceles triangle has 1 line of symmetry. It fits on itself only after the full turn, so its order is 1.

    • If you chose A: A square has 4 lines of symmetry and also turning symmetry of order 4. Go to M3.5.7.
    • If you chose C: A parallelogram has turning symmetry of order 2 but no line of symmetry. This question asks the other way round. Go to M3.5.7.

  6. A regular polygon has 6 sides. What is its order of turning symmetry?

    See the answer

    6

    A regular polygon fits on itself once for each of its corners in a full turn. 6 sides give order 6.

  7. From an earlier chapter

    The point (2, 3) is reflected in the mirror line x = 6. Where is the image?

    See the answer

    (10, 3)

    The point is 6 − 2 = 4 squares left of the mirror line. The image is 4 squares right, at 6 + 4 = 10. The height stays 3.

Keep it

Copy this into your exercise book.

The order of turning symmetry is how many times a shape fits on itself in one full turn. The smallest turn is 360° ÷ order. A pattern of order 4 copies a shape every 90°.

Draw: A square with a dot at its centre. Mark one corner 1, 2, 3 and 4 as it turns a quarter turn each time.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (801 KB). It opens on any phone.