Line Symmetry

A shape has line symmetry when it can be folded along a line so that both halves match exactly. Learn to identify axes of symmetry in common shapes, regular polygons, and everyday letters.

What Is Line Symmetry?

A shape has line symmetry (also called axial symmetry or mirror symmetry) when you can draw a straight line through it so that one half is the exact mirror image of the other.

The straight line is called the axis of symmetry (or line of symmetry).

Tip

Imagine folding the shape along the axis. If both halves fit together perfectly, the shape has line symmetry along that fold.

How to Find an Axis of Symmetry

1
Draw a line through the shape
Choose a line that might divide the shape into two equal parts — horizontal, vertical, or diagonal.
2
Reflect one half onto the other
Check whether every point on one side has a matching point at the same distance on the other side.
3
Decide
If the halves match perfectly, that line is an axis of symmetry. If not, try a different line.

Number of Axes in Common Shapes

Shape	Number of axes of symmetry
Circle	Infinitely many
Square	4
Rectangle	2
Equilateral triangle	3
Isosceles triangle	1
Scalene triangle	0
Regular pentagon	5
Regular hexagon	6

Axes of a Regular Polygon

n \text{ sides} \;\Rightarrow\; n \text{ axes of symmetry}

A regular polygon with n sides has exactly n axes of symmetry — one through each vertex and the midpoint of the opposite side.

Axes of Symmetry in Letters

Many capital letters have line symmetry:

Vertical axis only: A, M, T, U, V, W, Y
Horizontal axis only: B, C, D, E, K
Both vertical and horizontal axes: H, I, O, X

How many axes of symmetry does a square have?

A square has 4 axes of symmetry:

2 axes connecting the midpoints of opposite sides (one horizontal, one vertical)
2 diagonal axes connecting opposite corners

Each axis divides the square into two congruent mirror-image halves.

Real world: butterflies

A butterfly's body acts as the axis of symmetry — the left wing is the mirror image of the right wing. Scientists use this line symmetry when identifying and classifying butterfly species.

Real world: flags and architecture

Many national flags (Germany, France, Italy) have vertical line symmetry. The Taj Mahal has a central vertical axis of symmetry — its left half is the mirror image of its right half, making it one of the most famous examples of symmetry in architecture.

Drawing the Mirror Image of a Shape

1
Identify the key points
Choose the corners or important vertices of the shape you want to reflect.
2
Measure the perpendicular distance to the axis
Count grid squares (or use a ruler) to find how far each point is from the axis, measured at right angles.
3
Plot the mirror point on the opposite side
Place the new point at the same perpendicular distance on the other side of the axis.
4
Connect the mirror points
Join the reflected points in the correct order to complete the mirror image.

Common Mistakes

Measuring along the axis instead of perpendicular to it — always measure at right angles to the axis.
Stretching or shrinking the shape — the reflected image must be exactly the same size as the original.
Assuming every shape has at least one axis — scalene triangles and irregular shapes often have none.
Counting a shape's axes wrong — only count lines along which the halves truly match, not just any line through the centre.

Practice Problems

How many axes of symmetry does a regular hexagon have?

Show answer

6 axes of symmetry — one through each pair of opposite vertices and one through each pair of opposite edge midpoints.

A point P is 4 cm to the right of a vertical axis of symmetry. Where is its mirror image P′?

Show answer

P′ is 4 cm to the LEFT of the axis, at the same height as P.

Which of these letters has both a horizontal and a vertical axis of symmetry: A, H, B, F?

Show answer

H — it has both a vertical axis (left half mirrors right half) and a horizontal axis (top half mirrors bottom half).