Symmetry in three-dimensional shapes
🎯 In this topic you will
- Identify reflective symmetry in 3D shapes.
🧠 Key Words
- isometric paper
- plane of symmetry
Show Definitions
- isometric paper: A type of grid paper with equally spaced triangular lines used to draw 3D shapes accurately.
- plane of symmetry: An imaginary flat surface that divides a 3D shape into two identical mirror-image halves.
✨ Planes of Symmetry in 3D Shapes
Three-dimensional shapes can be symmetrical. For example, this chair is symmetrical. Instead of a line of symmetry, a 3D shape has a plane of symmetry. In three dimensions, a plane of symmetry divides a solid into two congruent parts.
❓ EXERCISES
1. These shapes are drawn on isometric paper.
Each shape has one plane of symmetry.
Copy the diagrams and draw the plane of symmetry on each shape.
👀 Show answer
Shape b has a vertical plane of symmetry.
Shape c has a horizontal plane of symmetry.
❓ EXERCISES
2. Shape a has two planes of symmetry. Shape b has three planes of symmetry.
Copy the diagrams and draw the planes of symmetry on each shape.
👀 Show answer
Shape b has two vertical planes and one horizontal plane of symmetry.
3. The diagram shows an L-shaped prism.
a. Draw the object on isometric paper.
b. The prism has one plane of symmetry. Draw the plane of symmetry on your isometric drawing.
c. Describe the plane of symmetry.
👀 Show answer
b. The plane of symmetry is drawn through the middle of the prism.
c. The plane of symmetry is vertical.
🧠 Think like a Mathematician
Task: Investigate the planes of symmetry in a cube.
Steps:
- Draw a cube. Draw a plane of symmetry that passes through four edges but no vertices.
- Draw the cube again. Draw a plane of symmetry that passes through four vertices of the cube.
- Copy and complete this statement: A cube has a total of ____ planes of symmetry.
- Draw diagrams to justify your answer to part c.
- Reflect on your answers and diagrams for parts a to d.
Follow-up Questions:
👀 Show Answer
- a: A vertical plane of symmetry that cuts the cube through the midpoints of four opposite edges.
- b: This plane of symmetry passes through four opposite vertices instead of edges.
- c: A cube has a total of $9$ planes of symmetry.
- d: Because all faces and dimensions of the cube are equal, the cube can be divided into two congruent mirror-image halves in multiple ways.
❓ EXERCISES
5. The diagram shows a 3D shape.
a. Describe the planes of symmetry for this shape.
b. Copy the shape and draw on the planes of symmetry.
👀 Show answer
b. When copied, the symmetry planes should be shown passing through the center vertically and horizontally, dividing the shape into congruent halves.
🧠 Think like a Mathematician
Investigation: Explore the connection between the lines of symmetry of 2D regular polygons and the planes of symmetry of the corresponding 3D prisms.
Task:
- Copy and complete the table below for the given polygons and prisms.
- Identify the relationship between the number of lines of symmetry of a polygon and the number of planes of symmetry of its prism.
- Use your observation to predict the number of planes of symmetry in a decagonal and dodecagonal prism.
- Reflect on and justify your answers with diagrams where possible.
| 2D regular polygon | Number of lines of symmetry | 3D prism | Number of planes of symmetry |
|---|---|---|---|
| Triangle | 3 | Triangular prism | 3 |
| Square | 4 | Square prism | 4 |
| Pentagon | 5 | Pentagonal prism | 5 |
| Hexagon | 6 | Hexagonal prism | 6 |
| Octagon | 8 | Octagonal prism | 8 |
Follow-up Questions:
i) a regular decagonal prism
ii) a regular dodecagonal prism
👀 Show Answer
- b: The number of planes of symmetry of the prism is the same as the number of lines of symmetry of its 2D polygonal base.
- c: i) A regular decagonal prism has $10$ planes of symmetry. ii) A regular dodecagonal prism has $12$ planes of symmetry.
- d: Each plane of symmetry of the polygon extends through the prism, dividing it into two identical halves. This explains why the counts are equal.
❓ EXERCISES
7.
a. Draw a cylinder.
b. Draw a plane of symmetry that passes through the circular ends of the cylinder.
c. Draw a plane of symmetry that does not pass through the circular ends of the cylinder.
d. How many planes of symmetry does a cylinder have? Explain your answer.
👀 Show answer
b. A vertical plane passing through the axis of the cylinder and the circular ends is a plane of symmetry.
c. A horizontal plane cutting the cylinder halfway between its two circular ends is also a plane of symmetry.
d. A cylinder has an infinite number of planes of symmetry passing through its central axis, plus one horizontal plane across its middle.