Symmetrical shapes and rotational symmetry

In a nutshell

Shapes that can be divided into two or more equal parts are symmetrical. The line that divides the shape into two equal parts is called the line of symmetry. Some shapes may have more than one line of symmetry, such as a rectangle. A shape has rotational symmetry when it looks the same in more than one position when rotated about its centre, the number of rotations is the order of rotational symmetry.

Symmetrical shapes

Symmetrical shapes are shapes that can be divided into two or more equal parts, down a dividing line of symmetry that passes through the centre of the shape. Lines of symmetry are vertical if they run down the centre of the image and horizontal if they run across the centre of the image. Look at the shapes in the images below, and think about how they may each be divided into equal parts by one or more dividing lines.

The shape below is a heart, and it has one vertical line of symmetry down its centre as shown. A love heart is therefore considered to be a symmetrical shape!

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

The shape below is an arrow, and it has one horizontal line of symmetry through its centre. An arrow is therefore considered to be a symmetrical shape!

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

On the other hand, a lightning bolt below is not a symmetrical shape, as there is no way to divide it into two equal parts.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

To check if a dividing line is a line of symmetry, it can help to imagine a mirror positioned down the line. If the shape is symmetrical, you would see the same shape in the reflection as you would without the mirror in place.

Example 1

How many lines of symmetry does a rectangle have?

Draw a rectangle and think about how the shape can be divided into equal parts with a mirror line.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Therefore, the rectangle has $\underline{2}$ lines of symmetry.

Example 2

How many lines of symmetry does an isosceles triangle have?

Draw an isosceles triangle and think about how the shape can be divided into equal parts with a mirror line.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Therefore, the triangle has $\underline{1}$ line of symmetry.

Example 3

How many lines of symmetry does a parallelogram have?

Draw a parallelogram and think about how the shape may be divided into equal parts with a mirror line.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

As you can see, there are no possible ways to divide this shape into equal parts.

Therefore, the parallelogram has $\underline{0}$ lines of symmetry.

Rotational symmetry

A shape has rotational symmetry when it looks the same in more than one position when rotated about its centre. The degree of rotational symmetry is determined by the number of orientations in which the shape looks the same as it did before rotation.

Example 4

What is the order of rotational symmetry of the shape below?

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Focus on one point and see how many times the shape can rotate to look the same before returning to the original position.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Therefore, the order of rotational symmetry of the shape is $\underline{4}$ .

Example 5

How many lines of symmetry does a five pointed star have? Does the star have rotational symmetry?

First, draw a five pointed star and think about all the different ways the shape can be divided into equal parts with a mirror line.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Next, draw a five pointed star and think about all the different ways the shape can be rotated to look the same, before returning to the original position.

Maths; Properties of shapes; KS3 Year 7; Symmetrical shapes and rotational symmetry

Therefore, the star has $\underline{5}$ lines of symmetry, and a rotational symmetry of order $\underline{5}$ .