Symmetry Revision Notes

A line of symmetry divides a shape into two parts that are mirror images of each other. If you fold the shape along this line, the two halves match exactly. The number of such lines varies depending on the shape's properties.

For example, an equilateral triangle has three lines of symmetry, each passing through a vertex and the midpoint of the opposite side. A square has four lines of symmetry: two along the diagonals and two through the midpoints of opposite sides.

To identify lines of symmetry, imagine folding the shape or use tracing paper to test if halves coincide. Irregular shapes often have no lines of symmetry.

• Lines of symmetry create mirror-image halves.
• The number of lines depends on the shape's regularity and side lengths.
• Equilateral triangles have 3 lines; squares have 4; rectangles have 2; scalene triangles have none.

Rotational symmetry occurs when a shape looks the same after being rotated about its centre by a certain angle less than 360°. The order of rotational symmetry is the number of times the shape matches itself during a full 360° rotation.

For example, an equilateral triangle has rotational symmetry of order 3 because it matches itself after rotations of 120°, 240°, and 360°. A square has order 4, matching itself every 90° rotation.

To find the order, divide 360° by the smallest angle of rotation that maps the shape onto itself. Shapes with no rotational symmetry other than 360° have order 1.

• Order of rotational symmetry = number of matching positions in 360° rotation.
• Equilateral triangle: order 3 (rotations of 120°).
• Square: order 4 (rotations of 90°).
• Irregular shapes often have order 1 (only at 360°).

In three dimensions, symmetry is described by planes of symmetry and axes of symmetry. A plane of symmetry divides a solid into two mirror-image halves, like a line of symmetry in 2D. An axis of symmetry is a line about which the solid can be rotated to look the same.

For example, a cylinder has an infinite number of planes of symmetry passing through its central axis, and infinite rotational symmetry about this axis. A regular triangular prism has planes of symmetry through the rectangular faces and axes of symmetry corresponding to rotations around its length.

Identifying these symmetries involves visualising or physically folding or rotating models.

• Plane of symmetry divides 3D shape into mirror halves.
• Axis of symmetry is a line about which rotation maps the shape onto itself.
• Cylinders have infinite planes and rotational symmetry about their axis.
• Prisms have planes through faces and rotational axes along their length.

Prisms: A regular prism has planes of symmetry through its faces and axes of rotational symmetry along its length. For example, a triangular prism has three planes of symmetry and rotational symmetry of order 3 about its axis.

Cylinders: Have infinite planes of symmetry through the axis and infinite rotational symmetry about the axis because any rotation maps the shape onto itself.

Pyramids: The number of planes of symmetry depends on the base. A square pyramid has four planes of symmetry through the apex and each base edge, and rotational symmetry of order 4 about the vertical axis if the base is regular and apex is aligned centrally. A triangular pyramid (tetrahedron) has fewer planes depending on its regularity and shape of faces and vertices, often 6 planes for a regular tetrahedron, with rotational symmetry of order 3 about certain axes.

• Prisms: planes through faces, rotational symmetry along length.
• Cylinders: infinite planes and rotational symmetry about axis.
• Pyramids: planes depend on base shape; rotational symmetry depends on apex alignment.
• Cones: one plane of symmetry through axis, infinite rotational symmetry about axis.

Recognising symmetry helps simplify geometric reasoning and calculations. For example, knowing the lines of symmetry can help find missing angles or lengths by reflecting parts of the shape.

In three dimensions, symmetry can aid in calculating surface areas or volumes by reducing the complexity of the shape. It also helps in visualising cross-sections and understanding properties like balance and centre of mass.

When solving problems, always check for symmetry to exploit these properties for efficient solutions.

• Use symmetry to find equal angles and lengths.
• Symmetry reduces complexity in 3D calculations.
• Check for symmetry before attempting constructions or proofs.
• Symmetry aids in classification and understanding shape properties.