Symmetry
Line and rotational symmetry
In this note
- 4.5Find the number of lines of symmetry of a shape or diagram.
- 4.5Find the order of rotational symmetry.
- 4.5Know the symmetries of triangles, quadrilaterals and regular polygons, and complete a diagram on a grid.
Next: Symmetry of solids.
Formula bank
Line symmetry
A line of symmetry splits a shape into two mirror-image halves: fold along it and they fit exactly. A rectangle has 2 - through the middles of its sides. Its diagonals are NOT lines of symmetry.
Rotational symmetry
A shape has rotational symmetry if it looks the same after a turn of less than 360° about its centre. The order is how many times it matches during one full turn. A parallelogram matches at 180° and 360°: order 2, but 0 lines.
| Shape | Lines | Order |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Parallelogram | 0 | 2 |
| Kite | 1 | 1 |
| Isosceles trapezium | 1 | 1 |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Regular n-sided polygon | n | n |
General methodHow to find the symmetry of a shape
- Lines: imagine folding along each possible line (vertical, horizontal, diagonals, through corners). Count those where the halves match.
- Order: imagine turning the shape about its centre; count how many times it looks the same before it is back where it started (including the start).
- Check against the table for special shapes; regular polygons: n and n.
Try it yourself
Show answer
Answer 1
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Answer 2
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Answer 5
Completing a diagram on a grid
General methodHow to shade squares to give a symmetry
- Line symmetry: draw the mirror line; reflect each shaded square across it; shade any reflection that is not shaded yet.
- Rotational symmetry of order 2: turn the grid upside down (half turn about the centre); shade where each square lands.
- Check the final diagram has exactly the symmetry asked for (no more, no less).
Example Shade one more square so the diagram has exactly one line of symmetry.
Try the vertical line: the left square on row 2 has no partner.
- Reflect in the vertical centre line
- shade row 2, last column
Answer one vertical line of symmetry
Example 2 Shade one more square so the diagram has rotational symmetry of order 2.
Half turn: each square must land on a shaded square.
- the square at row 1, column 2 lands on row 4, column 3
- shade row 4, column 3
Answer order 2, no lines
Ways they test it
Type 1 Shade squares
"Shade one more small square so the diagram has one line of symmetry." (1 mark)
Spot it a grid with some squares shaded.
- Pick the mirror line that needs only one more square.
- Shade the missing reflection.
Type 2 Order of rotational symmetry
"Write down the order of rotational symmetry of this shape." (1 mark)
Spot it 'order'.
- Count the matches in a full turn; minimum 1.
Type 3 Name the shape
"A quadrilateral has rotational symmetry of order 2 and no lines of symmetry. Name it." (1 mark)
Spot it symmetry clues.
- Use the table.
For example order 2, no lines
Answer parallelogram
Mistakes to avoid
- ✗ A rectangle's diagonals are lines of symmetry✓ only the two mid-linesFolding along a diagonal doesn't match.
- ✗ Order 0✓ order 1Every shape matches after a full turn.
- ✗ A parallelogram has 2 lines✓ 0 lines, order 2Turning is not folding.
Summary
- Line symmetry: fold so the halves match.
- Order of rotational symmetry: matches in one full turn (at least 1).
- Rectangle 2 & 2; rhombus 2 & 2; parallelogram 0 & 2; kite 1 & 1.
- Regular n-gon: n lines, order n.
If you see …
- "shade one more square"draw the mirror line or turn the grid, then fill the gap.
- "order of rotational symmetry"count the matches in one turn.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 4
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Answer 2
Show answer
Answer 3
Medium
Guide me
What is it asking? a name
What do we know? order 2: rectangle, rhombus, parallelogram
Plan only one has 0 lines
Show working
- parallelogram
Answer parallelogram
Guide me
What is it asking? a number
What do we know? regular
Plan n = 8
Show working
- 8
Answer 8
Guide me
What is it asking? a name
What do we know? one line of symmetry: kite or isosceles trapezium
Plan diagonals at 90°
Show working
- kite
Answer kite
Hard
Guide me
What is it asking? a number
What do we know? test vertical, horizontal, diagonals
Plan none fold
Show working
- 0 lines
Answer 0
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 0 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? an order
What do we know? half turn: matches
Plan quarter turn: doesn't
Show working
- order 2
Answer 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 2 | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? a number
What do we know? both diagonals
Plan fold along each
Show working
- 2 (both diagonals)
Answer 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 2 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can find lines of symmetry of shapes and diagrams.
- I can give the order of rotational symmetry.
- I can complete a diagram on a grid to have a given symmetry.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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