Symmetry - KS3 Maths - BBC Bitesize

Key points

An image of a symmetrical butterfly. A vertical dashed line of symmetry has been drawn passing through the centre. — Image caption,
Reflective symmetry can be seen in nature, such as with the wings of a butterfly.

To understand symmetry a good understanding of shapes, including regular polygons and triangles, can be helpful.
There are two types of symmetry: reflective and rotational.
If a shape does not have symmetry it is called asymmetrical.

Reflective symmetry

A is an imaginary line which splits a shape equally in two.

Each half must be a mirror image of the other.

If a can be folded in half, with either side of the fold being a mirror image of the other, then it has a line of symmetry.

A mirror can be also used to help find lines of symmetry. When a mirror is placed in a position on a shape and the reflection matches the other half of the shape, then this is a line of symmetry.

Examples

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By exploring the different ways of folding a rectangle, we can work out the number of lines of symmetry.
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The rectangle can be folded vertically down the middle. Both sides of the fold are the mirror image of each other. This fold is a line of symmetry.
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The rectangle can also be folded horizontally down the middle. Both sides of the fold are the mirror image of each other. This fold is a line of symmetry.
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In total, a rectangle has two lines of symmetry. It is not necessary to create a shape to check for lines of symmetry. The shape and its lines of symmetry could be sketched.
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Although the diagonal of a rectangle cuts the shape in half, it is not a line of symmetry.
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The fold shows the two sides are not the same. Using a mirror also confirms that the two halves are not identical.
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A square has four lines of symmetry: one vertical, one horizontal and two across the diagonals.
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Line symmetry doesn’t only apply to mathematical shapes. The no entry road sign has two lines of symmetry, one vertical and one horizontal.
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The letter ‘C’ has one horizontal line of symmetry.

Question

How many lines of symmetry does an isosceles triangle have?

An image of an isosceles triangle. The isosceles triangle is coloured pink.

Rotational symmetry

A shape has if it looks the same in more than one position when the shape is rotated about its centre.

The number of times it looks the same through a full turn (360°) is called the order of rotational symmetry. For example, a shape that looks the same five times through a full turn has a rotational symmetry of order five.

To find the order of rotational symmetry:

Choose a on the shape, usually a (corner).
Rotate the shape about its centre until the shape looks the same compared to its original starting position.
Continue to rotate the shape, counting how many times it looks identical to its starting position, until the shape has been rotated a full turn (360°).

Examples

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To work out the order of rotational symmetry of a rectangle, pick a reference point and label it A.
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Slowly rotate the shape clockwise about the centre, in the same direction as the hands on a clock. After a quarter turn (90°), reference point A is not in line with the original shape.
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After one half of a full turn (180°), reference point A is in line with the bottom left corner of the original shape. This is the first time the shapes look the same.
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After a further half of a turn (360° in total), the shape has returned to its original position. Reference point A is in line with the top right corner of the original shape. This is the second time the shapes look the same.
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The rectangle looked identical twice during the full turn, with reference point A at positions 1 and 2. The rectangle has rotational symmetry of order two.
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Symbols and logos often have rotational symmetry. To work out the order of rotational symmetry of this symbol, pick a reference point and label it A.
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After one third of a full turn (120°), the reference point A reaches position 1 where the shape looks the same as the original. After another third of a turn (120°), the reference point A reaches position 2 where the shape looks the same as the original. After the final third of a turn (120°), the reference point A reaches position 3 where the shape has returned to its original starting point. The recyclable symbol has rotational symmetry of order three.
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A regular pentagon has rotational symmetry of order five. Through a full turn (360°) there are five positions where reference point A would make the shape look the same as the original.
$Example four. A series of two images. The first image is a trapezium with two right angle vertices in the top left and bottom left. The top right vertex has been labelled as reference point A. Drawn right: a curved orange clockwise arrow representing a fraction of a turn. The second image is the same trapezium. The top right vertex has been labelled as one. The number is coloured orange. The trapezium is coloured blue.$
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Irregular shapes only look the same once through a full turn. There is only one position where reference point A would make the shape look the same as the original. Although this is called rotational symmetry of order one, the shape is described as having no rotational symmetry. Since this shape also has no line symmetry, it can be called asymmetrical.

Question

What is the order of rotational symmetry for an equilateral triangle?

Symmetry in regular polygons

A regular polygon has sides which are all the same length and interior angles which are all equal size.

In a regular polygon, the number of lines of symmetry is the same as the number of sides.

The order of rotational symmetry is same as the number of sides.

Examples

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A regular hexagon has six sides. A regular polygon has the same number of lines of symmetry as the number of sides. Therefore, a regular hexagon has six lines of symmetry. Three lines of symmetry pass through opposite vertices and three lines of symmetry pass through the mid-points of opposite sides.
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A regular hexagon has rotational symmetry of order six. Through a full turn (360°), there are six positions where reference point A would make the shape look the same as the original.
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A regular heptagon is a polygon with seven sides. Therefore, it has seven lines of symmetry. Each line of symmetry passes through a vertex (corner) to the midpoint of the opposite side.
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The regular heptagon has rotational symmetry of order seven. Through a full turn (360°), there are seven positions where reference point A would make the shape look the same as the original.
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A regular octagon is a polygon with eight sides. Therefore, it has eight lines of symmetry. Four lines of symmetry pass through opposite vertices and four lines of symmetry pass through the mid-points of opposite sides.
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The regular octagon has rotational symmetry of order eight. Through a full turn (360°), there are eight positions where reference point A would make the shape look the same as the original.

Question

How many lines of symmetry would a regular decagon, a 10-sided polygon, have?

Practise identifying symmetry

Quiz

Practise identifying symmetry with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of the London Eye. — Image caption,
The London Eye has 32 lines of symmetry.

The London Eye is an example of a symmetrical structure.

It is 135 metres in height and has a diameter of 120 metres. 32 viewing capsules, which are also called pods, are equally spaced apart on the wheel. This means the structure has rotational symmetry of order 32.

The cables joining the edge of the wheel to the central point (known as the spindle) replicate the 32 lines of symmetry.