Symmetry Revision Notes

Line symmetry occurs when a shape can be divided by a straight line into two parts that are mirror images of each other. This dividing line is called the line of symmetry.

To identify a line of symmetry, imagine folding the shape along a line; if the two halves match exactly, that line is a line of symmetry.

For example, an equilateral triangle has three lines of symmetry, each passing through a vertex and the midpoint of the opposite side.

• Line symmetry means the shape looks identical on both sides of the line.
• The number of lines of symmetry varies with the shape and its properties.
• Symmetry lines can be vertical, horizontal, or diagonal depending on the shape.

Rotational symmetry means a shape can be rotated about its centre by a certain angle less than 360° and still look exactly the same as the original shape.

The order of rotational symmetry is the number of times the shape matches itself during a full 360° rotation.

For example, a square has rotational symmetry of order 4 because it looks the same after rotations of 90°, 180°, 270°, and 360° (full turn).

• Order of rotational symmetry counts how many positions the shape fits onto itself in one full turn.
• The angle of rotation for each match is 360° divided by the order.
• Shapes with rotational symmetry of order 1 only match themselves at 360°, meaning no rotational symmetry other than the full turn.

Different types of triangles have different symmetry properties based on their side lengths and angles.

An equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3 (120° rotations).

An isosceles triangle has 1 line of symmetry (through the vertex angle and midpoint of the base) and rotational symmetry of order 1 (no rotational symmetry except full turn).  An scalene triangle has no lines of symmetry and rotational symmetry order 1.

• Equilateral: 3 lines of symmetry, rotational order 3.
• Isosceles: 1 line of symmetry, rotational order 1.
• Scalene: 0 lines of symmetry, rotational order 1.

Quadrilaterals show a range of symmetry depending on their type.

A square has 4 lines of symmetry (2 diagonals, 1 vertical, 1 horizontal) and rotational symmetry order 4 (90° rotations).

A rectangle has 2 lines of symmetry (vertical and horizontal through midpoints) and rotational symmetry order 2 (180° rotations).  A rhombus has 2 lines of symmetry (the diagonals) and rotational symmetry order 2.  A parallelogram (not rectangle or rhombus) has no lines of symmetry and rotational symmetry order 1.  A kite has 1 line of symmetry and rotational symmetry order 1.

• Square: 4 lines, order 4 rotational symmetry.
• Rectangle: 2 lines, order 2 rotational symmetry.
• Rhombus: 2 lines, order 2 rotational symmetry.
• Parallelogram: 0 lines, order 1 rotational symmetry.
• Kite: 1 line, order 1 rotational symmetry.

Regular polygons have as many lines of symmetry as sides and rotational symmetry order equal to the number of sides.

For example, a regular pentagon has 5 lines of symmetry and rotational symmetry order 5.

Irregular polygons generally have fewer or no lines of symmetry and rotational symmetry order 1 unless they have special properties.

• Regular polygons: lines of symmetry = number of sides.
• Rotational symmetry order = number of sides for regular polygons.
• Irregular shapes usually have minimal or no symmetry.