Symmetry
Symmetry of solids
In this note
- 4.5Count the planes of symmetry of prisms, cylinders, pyramids and cones.
- 4.5Find the order of rotational symmetry about an axis of a solid.
Before you start: Line and rotational symmetry.
Formula bank
Planes of symmetry
A plane of symmetry slices a solid into two halves that are mirror images. Each line of symmetry of the end face (or base) gives one plane through the solid; a prism also has one plane across its middle.
| Solid | Planes |
|---|---|
| cuboid (all edges different) | 3 |
| cuboid with two square faces | 5 |
| cube | 9 |
| prism, regular n-gon end | n + 1 |
| pyramid, regular n-gon base | n |
| cylinder, cone | infinitely many |
General methodHow to count planes of symmetry
- Look at the end face (prism) or base (pyramid) and count its lines of symmetry: each gives a plane along the solid.
- Prism: add 1 for the plane across the middle (the two ends are identical). Pyramid: add nothing.
- Check special cases: a cube has 9; anything with a circular cross-section has infinitely many.
Example How many planes of symmetry does a prism with an equilateral-triangle cross-section have?
Triangle: 3 lines; plus one across the middle.
- 3 + 1
Answer 4
Try it yourself
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Answer 7
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Answer 4
Axes of rotational symmetry
Turn the solid about an axis (a line through it): the order is how many times it fits onto itself in one turn. A square-based pyramid about its axis: order 4. A regular hexagonal prism about its long axis: order 6.
Try it yourself
Show answer
Answer 5
Ways they test it
Type 1 Count planes
"Write down the number of planes of symmetry of this prism." (1 mark)
Spot it a drawing of a solid.
- End face's lines of symmetry (+1 for a prism).
Type 2 Order about an axis
"Write down the order of rotational symmetry about the axis shown." (1 mark)
Spot it a dashed axis on the drawing.
- Same as the order of the end face or base.
Mistakes to avoid
- ✗ cuboid: 6 planes✓ 3Diagonal slices are not mirror planes.
- ✗ pyramid: n + 1 planes✓ nThere is no plane across the middle of a pyramid.
- ✗ cylinder: 2 planes✓ infinitely manyAny plane through the axis works.
Summary
- Each line of symmetry of the cross-section gives a plane.
- Prism: n + 1 planes; pyramid: n planes.
- Cuboid 3, cube 9, cylinder/cone infinitely many.
- Order about the axis = order of the end face or base.
If you see …
- a prismend face's lines + 1.
- a pyramidbase's lines only.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 3
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Answer 9
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Answer 4
Medium
Guide me
What is it asking? a number
What do we know? two square ends
Plan 3 + 2 diagonal planes
Show working
- 5
Answer 5
Guide me
What is it asking? a number
What do we know? hexagon: 6 lines
Plan no middle plane
Show working
- 6
Answer 6
Guide me
What is it asking? a number
What do we know? triangle: 3 lines
Plan + 1 across the middle
Show working
- 4
Answer 4
Hard
Guide me
What is it asking? a number
What do we know? octagon: 8 lines
Plan + 1
Show working
- 9
Answer 9
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 9 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? an order
What do we know? the octagon's order
Plan 8
Show working
- 8
Answer 8
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 8 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? a number
What do we know? no square faces
Plan 3
Show working
- 3
Answer 3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 3 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can count planes of symmetry of prisms, pyramids, cylinders and cones.
- I can give the order of rotational symmetry about an axis.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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