A move that changes nothing

In the last unit every transformation moved a shape somewhere new. Symmetry asks a different question: is there a transformation that leaves the shape looking exactly as it was?

Why anyone cares

Symmetry is a shortcut. If a shape has a line of symmetry, knowing one half tells you the other half for free, so you only have to measure, design or calculate half of it. A builder can draw half a house plan and mirror it.

Line symmetry

A line of symmetry is a line you could fold the shape along so that the two halves land exactly on top of each other. A shape can have none, one, two, several, or in the case of a circle, infinitely many.

Folding beats staring

Do not work out lines of symmetry by staring. Trace the shape, cut it out and fold it. The parallelogram is the classic trap: its diagonals look as though they should work, and they do not.

Rotational symmetry

A shape has rotational symmetry if you can turn it about its centre by less than a full turn and it looks exactly the same. The order is the number of times it looks the same during one complete turn. A square has order 4.

Order 1 means no rotational symmetry

Every shape looks the same after a full 360 degree turn, so that tells you nothing. A shape with order 1 is said to have no rotational symmetry. Rotational symmetry means order 2 or more.

The two kinds are independent

Shape	Lines of symmetry	Order of rotational symmetry
Square	4	4
Rectangle	2	2
Parallelogram	0	2
Kite	1	1 (none)
Equilateral triangle	3	3
Isosceles triangle	1	1 (none)
Regular hexagon	6	6
Circle	Infinitely many	Infinite

Two rows worth staring at

The parallelogram has no lines of symmetry, yet it has rotational symmetry of order 2. The kite is the opposite: one line of symmetry but no rotational symmetry. The two kinds of symmetry are genuinely separate ideas.

Regular polygons follow one rule

For a regular polygon with n sides there are exactly n lines of symmetry, and the order of rotational symmetry is also n. A regular pentagon has 5 of each, a regular octagon has 8 of each.

Where the lines actually sit

For an even number of sides, the lines either join opposite corners or the midpoints of opposite sides. For an odd number, every line runs from a corner to the midpoint of the opposite side, because there is never a corner directly opposite.

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Worked Example 1

Find the lines of symmetry and the order of rotational symmetry of a rectangle.

Try foldingTwo folds work: the vertical line and the horizontal line through the middle.

Now try the diagonals2 lines of symmetry. The diagonals look promising but fail the fold test.

Now turn itOrder 2. A 180 degree turn works, a 90 degree turn does not.

Worked Example 2

A shape has rotational symmetry of order 6. What is the smallest angle you can turn it through?

What order meansThe 6 positions are evenly spaced round the full 360 degrees.

Divide60 degrees, and it also works at 120, 180, 240, 300 and 360.

Read the question carefullyOrder is a count with no unit; angle is in degrees. Writing 6 degrees or order 60 loses the mark.

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Activity 9.1

Fold and turn: build a symmetry table for eight shapes

What you needPaper, scissors, a ruler and a drawing pin. Work in pairs so that one folds while the other records.

Method for lines of symmetryCut each shape out accurately and try every fold. A fold counts only if the halves match exactly.

Method for rotational symmetryDraw round the shape, pin it through its centre and turn it. Count how many times it fits the outline exactly in one full turn.

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Symmetry means a transformation leaves the shape looking exactly as it was.

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1What is a line of symmetry, and what is the most reliable way to find one?

2Explain what the order of rotational symmetry means, and why order 1 is said to be no rotational symmetry.

3Use the parallelogram and the kite to show that line symmetry and rotational symmetry are independent.

4How many lines of symmetry and what order of rotational symmetry does a regular polygon with n sides have? Explain why.

5Why is symmetry useful in practice, beyond making things look attractive?

Question 1 of 10

1A line of symmetry is a line along which

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Unit 9 · Symmetry

Symmetry

Mathematics · Grade 7

What You Will Be Able To Do

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Find every line of symmetry in a shape, and prove it by folding

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Work out the order of rotational symmetry and the smallest angle of turn

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Show that the two kinds of symmetry are completely independent

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Explain why symmetry saves an architect or an engineer half their work

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Last unit, every transformation moved a shape somewhere new. Symmetry asks a different question: is there a transformation that leaves the shape looking exactly as it was?

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Presenter notes: Open by connecting to the previous unit rather than starting fresh. Last week every transformation moved a shape somewhere new. Hold up a paper square, reflect it in a vertical line, and ask what changed. Nothing visible did, because the square landed exactly on itself. Then do the same with a paper letter F and it obviously changes. That contrast is the whole subject of this lesson: some shapes are unaffected by a transformation, and those are the symmetrical ones.