CIE IGCSE Mathematics (0580) Symmetry Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Symmetry Study Notes
LEARNING OBJECTIVE
- Symmetry
Key Concepts:
- Symmetry in 2D Solids
- Symmetry in 3D Solids
Symmetry
Symmetry
Symmetry describes how a shape or figure matches itself under certain transformations such as reflection or rotation.
Types of Symmetry:
- Line Symmetry (Mirror Symmetry): A shape has line symmetry if it can be folded along a line (called the line of symmetry) and both halves match exactly.
- Rotational Symmetry: A shape has rotational symmetry if it can be rotated about a point and still look the same at least once before a full turn (360°).
- Order of Rotational Symmetry: The number of times a shape matches itself during a 360° rotation.
Examples of Symmetry in Common Shapes:
Equilateral Triangle
- Has 3 lines of symmetry.
- Has order 3 rotational symmetry.
- All sides and all angles are equal.
Isosceles Triangle
- Has 1 line of symmetry (through the vertex angle).
- Has order 1 rotational symmetry (only looks the same once when rotated).
- Two equal sides and two equal angles.
Scalene Triangle
- No lines of symmetry.
- Order 1 rotational symmetry.
- All sides and angles are different.
Square
- Has 4 lines of symmetry.
- Has order 4 rotational symmetry.
- All sides are equal and all angles are 90°.
Rectangle
- Has 2 lines of symmetry (horizontal and vertical).
- Has order 2 rotational symmetry.
- Opposite sides are equal and all angles are 90°.
Rhombus
- Has 2 lines of symmetry (along diagonals).
- Has order 2 rotational symmetry.
- All sides are equal. Opposite angles are equal.
Parallelogram
- No lines of symmetry.
-
- Order 2 rotational symmetry.
- Opposite sides and angles are equal.
Regular Pentagon
- Has 5 lines of symmetry.
- Has order 5 rotational symmetry.
- All sides and angles are equal.
Circle
- Infinite lines of symmetry — any diameter is a line of symmetry.
- Infinite order of rotational symmetry.
- All points on the boundary are equidistant from the centre.
Example:
State the order of rotational symmetry of a regular hexagon.
▶️ Answer/Explanation
A regular hexagon looks the same after every 60° turn.
So the order of rotational symmetry is 6.
Example:
How many lines of symmetry does a rectangle have?
▶️ Answer/Explanation
A rectangle has 2 lines of symmetry: one vertical and one horizontal.
Symmetry in 3D Solids
Symmetry in 3D Solids
Symmetry in 3D shapes includes:
- Plane of Symmetry: A flat surface that divides a solid into two mirror-image halves.
- Axis of Symmetry (Rotational Symmetry): An imaginary line around which the shape can rotate and still look the same.
Symmetry in Common Solids:
1. Prisms
A prism has identical cross-sections along its length.
- Number of planes of symmetry depends on the base shape.
- Rectangular prism: 3 planes of symmetry and 3 axes.
2. Cylinders
- Infinite planes of symmetry through the vertical axis.
- 1 axis of symmetry (through the centre vertically).
3. Pyramids
Symmetry depends on the base shape.
- Square pyramid: 4 planes of symmetry and 1 vertical axis.
- Triangular pyramid (tetrahedron): 6 planes of symmetry.
4. Cones
- 1 plane of symmetry (vertical through the tip and base center).
- 1 axis of symmetry (vertical axis).
Example:
Which 3D shape has infinite planes of symmetry and one axis of symmetry?
▶️ Answer/Explanation
Answer: A cylinder
Reason: Any vertical plane through the center creates a mirror image, and it can rotate around its central axis.
Example:
A square-based pyramid has how many planes of symmetry?
▶️ Answer/Explanation
Answer: 4 planes of symmetry
Each plane cuts through the apex and one side or diagonal of the base.