### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Lines of Symmetry

A line of symmetry is an imaginary line that divides a shape into two equal halves, in a way that each half is a mirror image of the other. When a shape is folded along its line of symmetry, the two halves match exactly.

Symmetrical shapes have at least one line of symmetry, whereas asymmetrical shapes have none. Some examples of symmetrical shapes include:

• Squares – A square has four lines of symmetry: two diagonals and the vertical and horizontal lines that bisect the square.

• Rectangles – A rectangle has two lines of symmetry: one vertical and one horizontal line that bisect the rectangle.

• Equilateral triangles – An equilateral triangle has three lines of symmetry, which are the lines that bisect each angle and the opposite side.

• Circles – A circle has an infinite number of lines of symmetry. Since a circle is perfectly symmetrical, any line that passes through its centre will divide it into two equal halves that are mirror images of each other.

Each of these shapes has a unique number of lines of symmetry. In contrast, shapes like scalene triangles or parallelograms have no lines of reflection symmetry. This is because they cannot be divided into two mirror-image halves.

## Rotational Symmetry

Rotational symmetry occurs when a shape can be rotated around a central point and still look the same. The number of times a shape can be rotated within a full 360-degree rotation and still appear unchanged is called the order of rotational symmetry. For example:

A square has rotational symmetry of order 4. This is because it can be rotated four times (90°, 180°, 270° and 360°) and still look the same.

An equilateral triangle has rotational symmetry of order 3, as it can be rotated three times (120°, 240° and 360°) and still look the same.

A circle has infinite rotational symmetry, as it looks the same after any rotation.

Shapes like rectangles and scalene triangles do not have rotational symmetry. This is because they cannot be rotated and still appear the same within a full 360-degree rotation.

## Examples

How many lines of symmetry does an isosceles triangle have?

An isosceles triangle has one line of symmetry, and it goes through the midpoint of the base. How many lines of symmetry does a regular pentagon have?

A regular pentagon has five lines of symmetry, each passing through the centre of the pentagon and through opposite vertices. How many lines of symmetry does a regular heptagon have?

A regular heptagon has seven lines of symmetry, each passing through the centre of the heptagon and through opposite vertices. How many lines of symmetry does a parallelogram have?

A parallelogram has no lines of symmetry, as there is no way to divide it into two identical halves using a single straight line. 