Symmetry is a fundamental concept in geometry that refers to a sense of harmony and balance. When it comes to 3D shapes, understanding symmetry involves recognising planes of symmetry and rotational symmetry. This section explores the symmetry properties of various 3D shapes, such as cubes and cylinders, which are pivotal in the study of geometry at the IGCSE level.

Understanding Symmetry in 3D Shapes

Symmetry in 3D shapes extends the idea of line symmetry in two dimensions to three dimensions. A plane of symmetry divides a 3D object into two mirror-image halves. Rotational symmetry in 3D shapes refers to the shape looking the same after a certain amount of rotation about an axis.

Planes of Symmetry

A plane of symmetry divides a shape into two parts that are mirror images of each other.

Cubes

• Planes of Symmetry: A cube has 6 planes of symmetry. These planes can be identified as cutting through the centre, dividing the cube into mirrored halves.

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Cylinders

• Planes of Symmetry: A cylinder has an infinite number of planes of symmetry that pass through its axis.

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Rotational Symmetry

Rotational symmetry in 3D shapes involves rotation around an axis where the shape appears unchanged.

Example: Cube

• Axis through Faces: When rotated $90°, 180°, 270°, \text{or} 360°$ around an axis through the centres of opposite faces, a cube appears unchanged.
• Order of Rotational Symmetry: 4 (for each axis through the centres of opposite faces).

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Example: Cylinder

• Axis through the Centre: When rotated any degree around its longitudinal axis, a cylinder appears unchanged.
• Order of Rotational Symmetry: Infinite

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Practice Questions

Question 1: Sphere Symmetry

Determine the number of planes of symmetry and the order of rotational symmetry for a sphere.

Solution:

• Planes of Symmetry: Infinite, as any plane through the centre divides it into mirrored halves.
• Rotational Symmetry: Infinite, as it looks identical when rotated around any axis through its centre.

Question 2: Cube's Rotational Symmetry

Calculate the order of rotational symmetry of a cube around an axis through the centres of opposite edges.

Solution:

1. Identify axes through opposite edges.

2. Rotate the cube around one of these axes.

3. Observe at what angles the cube appears unchanged.

4. Rotational Symmetry: Order 2 (180° rotation returns cube to original appearance).