When one side of a shape is an exact mirror image of the other, this is called reflective symmetry. You can place a mirror across the middle of a symmetrical shape, and see the shape made whole again in the reflection. Another kind of symmetry is called rotational symmetry. If when a shape is rotated there are any points when the shape looks the same as it did originally, then it has rotational symmetry.

Line of symmetry

If you draw a line down the middle of a shape, and one half is an exact reflection of the other half, then the shape is symmetrical. This line is called the mirror line, or the line of symmetry. Mathematicians also call it the axis of symmetry.

Many shapes have more than one line of symmetry. A circle has infinite (an unlimited number of) lines of symmetry.

​​​​​​​If a shape or pattern has no lines of symmetry, we say it is asymmetric​​​​​​​.

Vertical or horizontal?

Vertical symmetry exists when the line of symmetry runs vertically (from top to bottom). The left and right sides are mirror images of one another.

Horizontal symmetry exists when the line of symmetry runs horizontally (from side to side). The top and bottom are mirror images of one another.

Polygons

Regular polygons always have the same number of lines of symmetry as the number of their sides.

Regular polygons are shapes in which all the sides are of equal length, and all the angles are equal.​​​​​​​

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Irregular polygons have sides of different lengths and angles of different sizes. They do not always have lines of symmetry, but some⁠—including kites, rectangles and isosceles triangles⁠—do.

Rotational symmetry

Imagine a pin placed through the centre of a shape enabling us to rotate the shape around this centre. If there are any points during a turn when the shape looks the same as it did originally, then it has rotational symmetry.

A four-pointed star has rotational symmetry of order 4. As we complete a full turn, there are 4 points at which it looks the same as the original shape.

If a shape never looks like itself during a full turn, then it has no rotational symmetry. Another way of saying this is to say it has rotational symmetry of order 1.


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Key words

full turn a complete turn through 360°

horizontal symmetry a type of symmetry in which the line of symmetry runs horizontally (from side to side)

line of symmetry a line drawn across a shape dividing it into two parts that exactly reflect one another

mirror image the image you would see reflected in a mirror

reflective symmetry a type of symmetry in which one side is the mirror image of the other

rotational symmetry if when a shape is rotated, there is more than one point at which it looks the same as the starting shape, we say it has rotational symmetry

symmetrical something is symmetrical when you draw a line down the middle and one side is an exact reflection of the other

vertical symmetry a type of symmetry in which the line of symmetry runs vertically (from top to bottom)