What is a line of symmetry?

A line of symmetry of a shape is a line which cuts the shape perfectly in half so that if you were to fold the shape along the line, both halves of the shape would overlap each other exactly. Equivalently, if you were to place a mirror along the line of symmetry, the shape would be unchanged. Here are some examples of lines of symmetry in nature:

Symmetry in shapes

Some shapes have no lines of symmetry, some have one and some have many! Regular shapes always have lines of symmetry, and the number of lines of symmetry is equal to the number of sides of the shape. Some irregular shapes have lines of symmetry, but the more irregular a shape is, the less likely it is to have a line of symmetry. Below is a guide on how to find lines of symmetry for a variety of shapes.

How to find a line of symmetry in symmetrical shapes

Children studying KS1 primary school maths will need to learn how to find vertical lines of symmetry in year 2. You may be shown a shape with a vertical line through it and asked whether or not the line is a line of symmetry. Alternatively, you may be given a shape and asked whether or not it has a vertical line of symmetry and if so to draw it on with a ruler. The diagram below shows some examples of shapes which are symmetrical, shapes which aren’t symmetrical and where the correct lines of symmetry should be drawn. Scroll down to the practice questions to have a go at some of these yourself. Once you’ve had a go, you can scroll down to the solutions to see if you got them right! These questions are good practice for any children who are going to sit the KS1 SATs.

Children studying KS2 primary school maths will build on their knowledge of symmetry from year 2 KS1 maths in year 3 and year 4. In year 4, Children should learn how to find all lines of symmetry of a 2D shapes as well as how to complete a symmetric figure where only part of the figure is drawn and the mirror line is shown. Scroll down to see where the lines of symmetry are in various different shapes and then have a go at the practice questions below. These questions are good practice for any children who are going to sit the KS2 SATs.

Lines of symmetry in a square

What is a line of symmetry in a square? Squares (and any other regular shapes with an equal number of sides) have two types of lines of symmetry:

• Ones which join opposite vertices together
• Ones which join the middle of one side of the shape to the middle of the opposite side.

Look at the image below to see where these are.

How many lines of symmetry does a square have?

A square is a regular quadrilateral which means that it has 4 lines of symmetry (2 of each of the types mentioned above). Here is a diagram so that you can see where they are:

Finding the line of symmetry of a pentagon

What is a line of symmetry in a pentagon? Regular pentagons (and any other regular shapes with an odd number of sides) only have one type of line of symmetry: lines which join the middle of one side of the shape to the opposite vertex.

How many lines of symmetry does a pentagon have?

A regular pentagon must have 5 lines of symmetry as regular shapes have equal numbers of lines of symmetry and sides. Here is a diagram so that you can see where they are:

Discover the line of symmetry of a rectangle

What is a line of symmetry in a rectangle? Whilst a square is a special kind of rectangle (i.e. one where all the side lengths are equal), let us now focus on rectangles where not all of the side lengths are equal.

We can think of a rectangle as a square which has been stretched to make two of the sides longer. Will all 4 of the lines of symmetry of a square still be lines of symmetry when the square has been stretched? The vertical and horizontal lines which join the centres of opposite sides still work. The lines which join opposite vertices of the rectangle will still cut the rectangle into 2 equally sized triangles, however they won’t overlap perfectly anymore when we fold along this line:

How many lines of symmetry does a rectangle have?

A rectangle with not all equal length sides therefore has 2 lines of symmetry. Here is a diagram so that you can see where they are:

Here’s how to find the line of symmetry in a circle

What is a line of symmetry in a circle? It turns out that any line you draw through a circle, as long as it goes through the centre, is a line of symmetry! This means it would be impossible to draw all of them because no matter how many we’ve drawn, we can always find another one.

How many lines of symmetry does a circle have?

Circles have an infinite number of lines of symmetry! Here is a diagram with some of them drawn on so that you can see how to draw them:

Do you know how to find the line of symmetry in a triangle?

What is a line of symmetry in a triangle? Any line of symmetry in a triangle joins a vertex of the triangle to the midpoint of the opposite side. How many of these there are depends on what kind of triangle you are dealing with: is it equilateral, isosceles or scalene?

How many lines of symmetry does a triangle have?

• Equilateral triangles have 3 equal angles and 3 equal sides and so are regular triangles. This means that they must have 3 lines of symmetry: join any vertex to the midpoint of the opposite side and it will be a line of symmetry.
• Isosceles triangles have 2 sides of equal length. If you join the vertex where the 2 equal sides meet to the midpoint of the opposite side, this will form a line of symmetry. No other lines work.
• Scalene triangles, i.e. triangles with 3 sides of different lengths, have no lines of symmetry.

The diagram below shows how to draw these lines of symmetry

Lines of symmetry in a parallelogram

A parallelogram is a quadrilateral with 2 pairs of parallel sides. What is a line of symmetry in a parallelogram? Where the lines of symmetry are and how many there are depends on what sort of parallelogram we are dealing with! Rectangles (and squares too!) are special parallelograms where all of the angles are right angles. As rectangles and squares have already been covered above, let’s look at parallelograms with different angles. (Note here that opposite angles in a parallelogram are equal).

Now let’s think about the side lengths. Opposite sides of a parallelogram are equal in length so we have 2 pairs of sides of equal length. What if we make all 4 sides equal? Then we get a special kind of parallelogram called a rhombus. Rhombi (this is just the plural of rhombus) have lines of symmetry joining opposite pairs of vertices. If we have 2 pairs of different sides however, there are no longer any lines of symmetry

How many lines of symmetry does a parallelogram have?

Special kinds of parallelograms such as squares, rectangles and rhombi have lines of symmetry as discussed above. Rhombi have 2 lines of symmetry as in the diagram below on the left. Generally though, parallelograms don’t have any lines of symmetry. Look at the diagram below on the right to see some failed attempts at finding a line of symmetry.

How symmetrical shapes work

For a shape to be symmetrical, it must have at least one line of symmetry. Any shape can be turned into a symmetrical shape by reflecting it in a mirror and adding the mirrored shape to the existing shape. This is a bit like painting half a butterfly on a piece of paper and then folding the piece of paper in half to make a picture of a whole symmetrical butterfly! Let’s have a look at an example of this. We’ll start with a parallelogram because we know that doesn’t have any lines of symmetry:

Symmetrical shapes – lines of symmetry practice questions

Here are some practice questions for your child to get more comfortable with how lines of symmetry work. They are split into 2 sections: the first is aimed at children in year 2 and year 3 who have learned about symmetry in KS1 maths and the second is aimed at children in year 4 and above who have in addition studied the symmetry topics in KS2 maths.

Lines of symmetry in maths questions for year 2 and above

The following questions are similar to those in KS1 SATs papers and are good practice for any children in year 2 who will be sitting these or older children trying to get a better understanding of symmetry.

Symmetry geometry questions for year 4 and above

The following questions are similar to those in KS2 SATs papers and are good practice for any children in year 4 and above who will be sitting these.

Lines of symmetry solutions

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