Sec 1 · G1 (N(T))

Symmetry

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1. Line symmetry

Lines of symmetry dashed: a square has 4, a rectangle 2, an equilateral triangle 3, a rhombus 2 (its diagonals), a parallelogram 0, a kite 1, an isosceles trapezium 1 and a regular hexagon 6.

Word	Meaning
line of symmetry	a fold line: the two halves match exactly
regular polygon, $n$ sides	$n$ lines of symmetry

📖 Worked example

The diagram shows a kite. State whether the dashed line is a line of symmetry of the shape.

Step 1 of 2

Imagine folding the shape along the dashed line

Each corner on one side must land exactly on a corner on the other side, and each side on a side.

📖 Worked example

The diagram shows a rectangle. Write down the number of lines of symmetry of the shape.

Step 1 of 2

Look for every fold line that makes the two halves match

Lines of symmetry: the two lines joining the middles of opposite sides (not the diagonals).

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2. Rotational symmetry

Each shape turned about its centre until a marked corner lands on the next corner: an equilateral triangle after 120° (order 3), a square after 90° (order 4), a parallelogram after 180° (order 2); a kite fits only after a full turn (order 1).

Order: how many times it fits in one full turn

\displaystyle \text{angle of rotation} = \dfrac{360^\circ}{\text{order}}

e.g.

\displaystyle \dfrac{360^\circ}{4}

\displaystyle = 90^\circ

a square: order $4$

Shape	Lines of symmetry	Order	Angle of rotation
square	$4$	$4$	$90^\circ$
rectangle	$2$	$2$	$180^\circ$
equilateral triangle	$3$	$3$	$120^\circ$
rhombus	$2$	$2$	$180^\circ$
parallelogram	$0$	$2$	$180^\circ$
kite	$1$	$1$
regular hexagon	$6$	$6$	$60^\circ$

📖 Worked example

The diagram shows a propeller shape. Write down its order of rotational symmetry.

Step 1 of 1

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

📖 Worked example

The diagram shows an equilateral triangle. Find the smallest angle through which it can be rotated about its centre to look exactly the same.

Step 1 of 2

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

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1. Line symmetry

Word	Meaning
line of symmetry	a fold line: the two halves match exactly
regular polygon, $n$ sides	$n$ lines of symmetry

📖 Worked example

The diagram shows a kite. State whether the dashed line is a line of symmetry of the shape.

Step 1 of 2

Imagine folding the shape along the dashed line

Each corner on one side must land exactly on a corner on the other side, and each side on a side.

📖 Worked example

The diagram shows a rectangle. Write down the number of lines of symmetry of the shape.

Step 1 of 2

Look for every fold line that makes the two halves match

Lines of symmetry: the two lines joining the middles of opposite sides (not the diagonals).

Practise this →

2. Rotational symmetry

Order: how many times it fits in one full turn

\displaystyle \text{angle of rotation} = \dfrac{360^\circ}{\text{order}}

e.g.

\displaystyle \dfrac{360^\circ}{4}

\displaystyle = 90^\circ

a square: order $4$

Shape	Lines of symmetry	Order	Angle of rotation
square	$4$	$4$	$90^\circ$
rectangle	$2$	$2$	$180^\circ$
equilateral triangle	$3$	$3$	$120^\circ$
rhombus	$2$	$2$	$180^\circ$
parallelogram	$0$	$2$	$180^\circ$
kite	$1$	$1$
regular hexagon	$6$	$6$	$60^\circ$

📖 Worked example

The diagram shows a propeller shape. Write down its order of rotational symmetry.

Step 1 of 1

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

📖 Worked example

The diagram shows an equilateral triangle. Find the smallest angle through which it can be rotated about its centre to look exactly the same.

Step 1 of 2

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

Practise this →