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Year 3 Maths 2D Shapes & Symmetry

Line symmetry

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Theory

Line symmetry is a Year 3 maths shape skill: a line of symmetry is a fold line that divides a shape into two matching halves that land exactly on top of each other. This page explains the fold test, how to count the lines of symmetry for squares, rectangles, triangles and regular shapes, and how to find a matching point across a mirror line. In the Australian Curriculum and the Victorian Curriculum this is Year 4 content; it is taught in Year 3 in Western Australia. Curriculum: VC2M4SP04 / AC9M4SP03.

A line of symmetry is a straight line that folds a shape into two matching halves. When you fold along it, one half lands exactly on top of the other. It is sometimes called a mirror line, because one half looks like the reflection of the other.

A shape has line symmetry if at least one fold like this works. Every point on one side has a matching point the same distance away on the other side of the line.

Different shapes have different numbers of lines. Some have just one, some have several, and some have none at all. A regular shape, with all sides and corners the same, has as many lines of symmetry as it has sides.

Three shapes with their lines of symmetry A square with 4 dashed lines of symmetry, an equilateral triangle with 3, and a rectangle with 2. Each dashed red line is a fold line where the two halves match. Square — 4 Triangle — 3 Rectangle — 2
The dashed red lines are the lines of symmetry. A regular shape has as many as it has sides.
Two matching halves across a mirror line A house-shaped figure split by a vertical dashed red mirror line. The left half and right half are mirror images. Point P on the left matches point P prime on the right, the same distance from the line. P P' mirror line
The two halves are mirror images. Point P matches P', the same distance across the line.

These are the numbers of lines of symmetry for the shapes you meet most often. They are worth remembering.

ShapeLines of symmetry
Square4
Rectangle (not a square)2
Equilateral triangle3
Isosceles triangle1
Scalene triangle0
Regular pentagon5
Regular hexagon6
Slanted parallelogram0
Circlemore than you can count

For any regular shape (all sides and corners the same), there is a quick rule:

\[ \text{lines of symmetry} = \text{number of sides} \]
lines of symmetry=number of sides
Regular shape rule. A regular shape with 10 equal sides and 10 equal corners has 10 lines of symmetry. Count the sides of a regular shape and you have your answer.

How to find the lines of symmetry of a shape

  1. Try the fold test. Imagine folding the shape along a line and check whether the two halves land exactly on top of each other.
  2. Test the likely lines. Try folding down the middle, across the middle, and along each diagonal. Each fold that works is one line of symmetry.
  3. Count them up. Add up every fold that works. If none of them match, the shape has 0 lines of symmetry.
Example 1 — A rhombus
A rhombus has four equal sides but no square corners. How many lines of symmetry does it have?
A rhombus with its lines of symmetry A rhombus standing on one corner. Two red dashed lines join opposite corners: one upright and one across. Both are lines of symmetry.
Solution

Fold along the line joining the top and bottom corners: the two halves match.

Fold along the line joining the left and right corners: the halves match again.

Folding down the middle of a side does not work, because the slanted sides do not land on each other. A rhombus has \(2\) lines of symmetry.

Example 2 — A capital letter
How many lines of symmetry does a plain capital letter M have?
Solution

Fold M down the middle: the left half lands exactly on the right half, so that fold works.

No across fold works, so M has \(1\) line of symmetry.

Example 3 — A square's diagonal
Is the dashed diagonal of this square a line of symmetry?
A square with one dashed diagonal A square with a single red dashed line from the top-left corner to the bottom-right corner.
Solution

Fold the square along the diagonal. The top-right corner lands exactly on the bottom-left corner, because all four sides are the same length.

The two halves match, so yes, the diagonal of a square is a line of symmetry.

Example 4 — Matching point
The dashed line is a line of symmetry. Point \(P\) is \(2\) squares to the left of the line. Where is its matching point?
A point and its matching point across a mirror line A square grid with a red dashed upright mirror line. Point P is 2 squares to the left of the line. Its matching point P prime is 2 squares to the right, in the same row. P P'
Solution

A matching point is the same distance from the line, on the other side, in the same row.

The matching point \(P'\) is \(2\) squares to the right of the line, level with \(P\).

Example 5 — Finish the figure
The left half of a symmetrical figure is drawn on a grid. It is \(3\) squares wide. How wide is the whole figure when it is finished?
Solution

The right half is a mirror image of the left half, so it is also \(3\) squares wide.

\(3 + 3 = 6\), so the whole figure is \(6\) squares wide.

Common pitfalls

A diagonal is not always a line of symmetry. Watch out for the diagonal of a rectangle. When you fold along it, the corners miss and the halves do not match, so it is not a line of symmetry.
Tidy does not mean symmetric. A slanted parallelogram looks neat, but no fold makes its halves match, so it has 0 lines of symmetry.
Same distance, not double. A matching point sits the same distance from the mirror line, not twice as far. If a point is 4 squares from the line, its match is 4 squares from the line too.

Frequently asked questions

What is a line of symmetry?

A line of symmetry is a fold line that splits a shape into two matching halves. When you fold along it, one half lands exactly on top of the other.

How do you find the lines of symmetry of a shape?

Imagine folding the shape along a line. If the two halves land exactly on top of each other, that line is a line of symmetry. Try folding down the middle, across the middle and along the diagonals.

Why is a square's diagonal a line of symmetry but a rectangle's is not?

All four sides of a square are equal, so folding along a diagonal puts each side on top of another side of the same length. A rectangle has long and short sides, so a diagonal fold makes the corners miss.

Do letters have lines of symmetry?

Many do. A plain capital X has \(2\) lines (down and across), a capital B has \(1\) line (across the middle) and a capital F has none.

What is a mirror line?

A mirror line is another name for a line of symmetry. If you stood a mirror on the line, the reflection would show the other half of the shape.

Does a line of symmetry have to go straight up and down?

No. A line of symmetry can go up and down, across, or on a slant, like the diagonals of a square.

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