Symmetry - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter explores symmetry in its different forms -- line, rotational, and reflection -- helping students recognise patterns in geometry, nature, and art.
1. Why this chapter matters
Symmetry is everywhere -- in butterfly wings, snowflakes, buildings, and even the human face. Understanding symmetry develops spatial reasoning and aesthetic sense. In CBSE exams, this chapter contributes 4-6 marks with diagram-based and identification questions.
2. Line symmetry (reflection symmetry)
A figure has line symmetry if it can be folded along a line so that the two halves match exactly.
The fold line is called the axis of symmetry or line of symmetry.
Examples of line symmetry
- A butterfly: 1 line of symmetry (vertical).
- A square: 4 lines of symmetry (two diagonals, horizontal, vertical).
- A rectangle: 2 lines of symmetry (horizontal and vertical).
- A circle: Infinite lines of symmetry.
- An equilateral triangle: 3 lines of symmetry.
- An isosceles triangle: 1 line of symmetry.
- A scalene triangle: 0 lines of symmetry.
3. Multiple lines of symmetry
| Shape | Number of lines of symmetry |
|---|---|
| Square | 4 |
| Rectangle | 2 |
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinite |
| Parallelogram | 0 |
A regular polygon with n sides has n lines of symmetry.
4. Rotational symmetry
A figure has rotational symmetry if it can be rotated (less than a full turn) about a point and look exactly the same as its original position.
Key terms
- Centre of rotation: The point about which the figure rotates.
- Angle of rotation: The smallest angle through which the figure is rotated to match its original position.
- Order of rotational symmetry: The number of times the figure looks the same in one complete rotation (360 degrees).
Finding order of rotational symmetry
Order = 360 degrees / Angle of rotation.
Examples
| Shape | Angle of rotation | Order of rotational symmetry |
|---|---|---|
| Equilateral triangle | 120 degrees | 3 |
| Square | 90 degrees | 4 |
| Rectangle | 180 degrees | 2 |
| Regular pentagon | 72 degrees | 5 |
| Circle | Any angle | Infinite |
| Parallelogram | 180 degrees | 2 |
| Scalene triangle | 360 degrees | 1 |
5. Comparison: line vs rotational symmetry
| Property | Line symmetry | Rotational symmetry |
|---|---|---|
| Transformation | Folding (reflection) | Rotation |
| Fixed element | Axis (line) | Point (centre) |
| Result | Two matching halves | Same appearance after rotation |
| Example | Butterfly | Windmill blades |
| Can exist independently | Yes | Yes |
Some figures have both line and rotational symmetry (e.g., square, equilateral triangle, regular hexagon). Some have only one type.
6. Reflection symmetry
Reflection symmetry is another name for line symmetry. A figure reflected across a line produces a mirror image.
Properties of reflection
- The image is the same distance from the mirror line as the original.
- The image is reversed (left becomes right).
- The image is congruent to the original.
7. Symmetry in letters and numbers
Letters with line symmetry
- A: 1 line (vertical)
- B: 1 line (horizontal)
- C: 1 line (horizontal)
- D: 1 line (horizontal)
- H: 2 lines (horizontal and vertical)
- I: 2 lines (horizontal and vertical)
- M: 1 line (vertical)
- O: Infinite (circle-like)
- T: 1 line (vertical)
- X: 2 lines (diagonals or horizontal-vertical)
Letters with rotational symmetry
- H: Order 2
- I: Order 2
- N: Order 2
- O: Order 2 (or infinite for circle)
- S: Order 2
- X: Order 2
- Z: Order 2
8. Symmetry in nature and design
Nature is full of symmetry: flowers (radial symmetry), leaves (bilateral symmetry), snowflakes (hexagonal symmetry), and starfish (pentagonal symmetry). Architects and designers use symmetry to create balanced and aesthetically pleasing structures.
9. Worked examples
Example 1: Draw all lines of symmetry for a regular hexagon.
A regular hexagon has 6 lines of symmetry: 3 lines joining opposite vertices and 3 lines joining midpoints of opposite sides.
Example 2: Find the order of rotational symmetry of a regular octagon.
Angle of rotation = 360/8 = 45 degrees. Order = 360/45 = 8.
Example 3: A figure has rotational symmetry of order 5. What is its angle of rotation?
Angle of rotation = 360/5 = 72 degrees.
Example 4: Does the letter 'S' have line symmetry? Does it have rotational symmetry?
S has no line symmetry (no fold line matches both halves). S has rotational symmetry of order 2 (looks same when rotated 180 degrees).
10. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Thinking all triangles have line symmetry | Only isosceles and equilateral triangles have line symmetry |
| Confusing order of rotation with number of lines | Order counts rotational matches; lines count fold lines |
| Thinking parallelogram has line symmetry | A parallelogram has rotational but NO line symmetry |
| Diagonals of rectangle as symmetry lines | Rectangle diagonals are NOT lines of symmetry |
| Counting 360 degree turn as rotational symmetry | Every figure has that. Look for SMALLER angles |
11. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Draw lines of symmetry | 2 marks | 1 question |
| Find order of rotational symmetry | 2 marks | 1 question |
| Identify symmetrical figures | 1 mark | 1-2 questions |
| Symmetry in letters/shapes | 2 marks | 1 question |
| Complete symmetrical figure | 3 marks | Occasional |
12. Self-test
- How many lines of symmetry does a regular pentagon have?
- Find the order of rotational symmetry of a rectangular sheet of paper.
- Does an isosceles trapezium have line symmetry? Rotational symmetry?
- Name a letter that has both line symmetry and rotational symmetry.
- What is the angle of rotation for a figure of rotational symmetry order 6?
- Complete the figure: A shape has half drawn. The other half is a mirror image across a vertical line. Draw the complete figure for: a right half showing a semi-circle and a triangle.
13. Answer key
- A regular pentagon has 5 lines of symmetry.
- Rectangle has rotational symmetry of order 2 (looks same after 180 degree rotation).
- An isosceles trapezium has 1 line of symmetry (vertical). It has no rotational symmetry (order 1).
- H, I, O, X have both line and rotational symmetry. (Accept any one.)
- Angle = 360/6 = 60 degrees.
- Draw the mirror image: the left half should be an exact reflection. The semi-circle and triangle should be mirrored across the vertical line.
14. Quick revision
- Line symmetry: two halves match after folding along a line.
- Regular polygon with n sides has n lines of symmetry.
- Rotational symmetry: figure matches original after rotation.
- Order of rotational symmetry = 360 / angle of rotation.
- Some figures have both types of symmetry.
- Reflection produces a mirror image congruent to the original.
- Parallelogram has rotational but NOT line symmetry.
