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Symmetry

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How to find the lines of symmetry and the order of rotational symmetry of shapes and letters, complete a shape or pattern on a grid, and use symmetry to find missing lengths.

Symmetry in the exam

A shape has symmetry when one part of it matches another part exactly. There are two kinds: line symmetry, where one half is the mirror image of the other, and rotational symmetry, where the shape looks the same after part of a turn.

Early in a paper, you may be asked to write down the number of lines of symmetry of a shape, or its order of rotational symmetry, for 1 mark each. You may also be asked to complete a shape or a pattern on a grid so that it has a line of symmetry.

Some questions ask you to draw a shape with given symmetry, such as a quadrilateral with no lines of symmetry and rotational symmetry of order 2. Later in a paper, a question worth 4 or 5 marks may tell you that a shape has one line of symmetry, and you use that to find a length or an angle.

Lines of symmetry

A line of symmetry splits a shape into two halves that are mirror images. If you folded the shape along the line, the two halves would fit exactly on top of each other. It is also called a mirror line, and it is usually drawn dashed.

A rectangle has 2 lines of symmetry. Its diagonals are not lines of symmetry. Fold a rectangle along a diagonal and the corners do not meet.

Worked example 1: Lines of symmetry of a rhombus

Here is a rhombus. How many lines of symmetry does the rhombus have?

  1. Try the line through the top and bottom corners. Fold along it, and the left half fits exactly on the right half. It is a line of symmetry.
  2. Try the line through the left and right corners. Fold along it, and the top half fits exactly on the bottom half. It is a line of symmetry too.
  3. Try a line through the middles of two opposite sides, as on a rectangle. Fold along it, and the corners do not meet, so it is not a line of symmetry.

  4. Your turn. How many lines of symmetry does the rhombus have?

    Only the two diagonals work, so the rhombus has 2 lines of symmetry.

  5. Answer: 2

Rotational symmetry

A shape has rotational symmetry if it looks exactly the same after less than a full turn about its centre. The order of rotational symmetry is the number of times it looks exactly the same during one full turn.

To find the order, trace the shape, turn the tracing about the centre, and count how many times it fits on the shape until you are back at the start. Every shape fits after a full turn, so the lowest order is 1. Order 1 means the shape has no rotational symmetry.

The smallest turn that makes a shape look the same is a full turn divided by the order:

\text{smallest turn}=360^\circ\div\text{order}

The windmill has order 4, so it turns through 360^\circ\div4=90^\circ, a quarter turn.

Worked example 2: Symmetry of a letter

Here is the capital letter N. (a) Write down the number of lines of symmetry of the letter N. (b) Write down the order of rotational symmetry of the letter N.

  1. (a) Try a line down the middle. In the mirror image the sloping stroke would go the other way, so the two halves do not match.
  2. A line across the middle does not work either: the top half is not the mirror image of the bottom half. The letter N has no lines of symmetry.
  3. (b) Turn the N half a turn, 180^\circ, about its centre. The left stroke moves to where the right stroke was, the sloping stroke lands on itself, and the N looks exactly the same.

  4. Your turn. After a quarter turn the N lies on its side, so it does not fit. In one full turn, how many times does the N look exactly the same?

    It fits after a half turn and again after a full turn, so the order of rotational symmetry is 2

  5. Answer: (a) 0 (b) 2

Quadrilaterals and regular polygons

  • Square: 4 lines of symmetry, order 4
  • Rectangle: 2 lines of symmetry, order 2
  • Rhombus: 2 lines of symmetry (its diagonals), order 2
  • Parallelogram: no lines of symmetry, order 2
  • Kite: 1 line of symmetry, order 1
  • Isosceles trapezium: 1 line of symmetry, order 1

So a quadrilateral with no lines of symmetry and rotational symmetry of order 2 is a parallelogram. A square, a rectangle, a rhombus or an isosceles trapezium will not do, because each of those has lines of symmetry.

A regular polygon has all its sides equal and all its angles equal. A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n. An equilateral triangle has 3 of each, and a regular hexagon has 6 of each.

Worked example 3: Symmetry of a regular polygon

Here is a regular polygon with 10 sides. (a) Write down the number of lines of symmetry of the polygon. (b) Write down its order of rotational symmetry. (c) Work out the smallest angle the polygon must turn through about its centre so that it looks exactly the same.

  1. (a) The polygon is regular, so it has as many lines of symmetry as it has sides: 10
  2. (b) Its order of rotational symmetry is also the number of sides: 10
  3. (c) Smallest turn =360^\circ\div\text{order}=360^\circ\div10

  4. Your turn. Work out 360\div10

    360\div10=36^\circ

  5. Answer: (a) 10 (b) 10 (c) 36^\circ

Completing a shape or pattern on a grid

To complete a shape so that a dashed line is a line of symmetry, find the partner of each corner. Count how many squares the corner is from the line. Then count the same number of squares on the other side, going straight across, at right angles to the line.

A corner that is on the line stays where it is: it is its own partner. When you have found every partner, join the corners in order with a ruler. Shaded squares work the same way: each one needs a partner the same distance from the line on the other side.

For rotational symmetry of order 2, think of a half turn about the centre. Each shaded square needs a partner the same distance from the centre, on the exact opposite side. Turning the page upside down is a good check.

Worked example 4: Completing a shape on a coordinate grid

The diagram shows half of a shape on a grid. The dashed line is a line of symmetry of the whole shape. Complete the shape, and write down the coordinates of the corner that matches C.

  1. A (1,4) and D (7,4) are on the dashed line, so they stay where they are.
  2. B (2,6) is 2 squares above the line. Its partner is 2 squares below the line, straight down: (2,2)

  3. Your turn. C is at (5,7). Write down the coordinates of its partner.

    Write your answer as (x, y)

    C is 3 squares above the line, so its partner is 3 squares below it: (5,1)

  4. Join D to (5,1), then to (2,2), then back to A.
  5. Answer: (5,1)

Worked example 5: Shading a pattern with rotational symmetry

Here is a pattern on a grid. Five squares, P, Q, R, S and T, are shaded. Work out the smallest number of extra squares that must be shaded so that the pattern has rotational symmetry of order 2.

  1. Turn the pattern half a turn about the dot at the centre. Every shaded square must land on a shaded square.
  2. P, in the top left corner, lands on the bottom right corner, which is T. T is shaded, so P and T already match.
  3. S, in the bottom left corner, lands on the top right corner. That square is not shaded, so it must be shaded: 1 extra square so far.

  4. Your turn. Now turn Q and R in the same way. How many extra squares must be shaded altogether?

    Q lands on the third square of the bottom row, and R on the third square of the second row up. Neither is shaded, so 1+2=3 extra squares.

  5. Answer: 3

Using symmetry to find lengths and angles

A line of symmetry splits a shape into two matching halves. So every side has a matching side of the same length on the other half, and every angle has a matching angle of the same size.

For example, if a pentagon has one line of symmetry and one angle on the left of the line is 108^\circ, the matching angle on the right is also 108^\circ. Write down which sides and angles match before you start.

In a perimeter question, add up every side, including the matching sides that have no label. Then make the total equal to the perimeter and solve the equation.

Worked example 6: Using symmetry in a perimeter problem

The diagram shows a hexagon PQRSTU. The dashed line is a line of symmetry of the hexagon. All lengths are in centimetres. The perimeter of the hexagon is 37 cm. Work out the length of QR.

  1. UP matches QR, so UP =x+3. TU matches RS, so TU =5
  2. Add all six sides: 3x+(x+3)+5+2x+5+(x+3)=37
  3. Collect like terms: 7x+16=37

  4. Your turn. Solve 7x+16=37. What is x?

    7x=37-16=21, so x=21\div7=3

  5. QR =x+3=3+3=6. Check: 9+6+5+6+5+6=37
  6. Answer: QR =6 cm

Common mistakes

  • Counting the diagonals of a rectangle as lines of symmetry. A rectangle has 2 lines of symmetry, through the middles of its sides; fold it along a diagonal and the corners do not meet.
  • Drawing a square, a rectangle or an isosceles trapezium for "no lines of symmetry, order 2". Each of these has lines of symmetry; the quadrilateral you want is a parallelogram.
  • Giving order 0 for a shape with no rotational symmetry. Every shape fits after a full turn, so the lowest order is 1.
  • Putting a reflected corner one square out. Count the squares from the corner to the line, then count the same number on the other side, straight across.
  • Leaving out the matching sides in a perimeter. Use the line of symmetry to find the sides with no label, then add all of them.

Practice questions

More on this topic: Symmetry worksheet with full solutions