Symmetry
1. What Is Symmetry?
SYMMETRY means BALANCE and PROPORTION. A figure is SYMMETRICAL if one half is the MIRROR IMAGE of the other half.
Types of Symmetry
| Type | Description |
|---|---|
| Line Symmetry (Reflection Symmetry) | A figure can be folded along a line so halves match exactly |
| Rotational Symmetry | A figure looks the same after a rotation of less than 360° |
| Point Symmetry | Every point has a matching point at equal distance from a centre |
2. Line Symmetry (Reflection Symmetry)
A figure has LINE SYMMETRY if there is a LINE (called the AXIS of symmetry) that divides it into two IDENTICAL halves.
Examples
- A SQUARE has 4 lines of symmetry.
- A RECTANGLE has 2 lines of symmetry.
- An EQUILATERAL TRIANGLE has 3 lines of symmetry.
- An ISOSCELES TRIANGLE has 1 line of symmetry.
- A CIRCLE has INFINITE lines of symmetry.
- A SCALENE TRIANGLE has NO line of symmetry.
Table: Lines of Symmetry for Common Shapes
| 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 |
| Rhombus | 2 |
Worked Example (ICSE 2024, 2 marks)
'Draw all lines of symmetry for a regular hexagon.'
Solution: A regular hexagon has 6 lines of symmetry — 3 through opposite vertices and 3 through midpoints of opposite sides.
3. Rotational Symmetry
A figure has ROTATIONAL SYMMETRY if it looks the SAME after being rotated by an angle LESS than 360°.
Key Terms
- Centre of rotation: The fixed point around which the figure is rotated.
- Angle of rotation: The smallest angle through which the figure is rotated to look the same.
- Order of rotational symmetry: The number of times the figure looks the SAME during a FULL 360° rotation.
Formula
Order of rotational symmetry = 360° / Angle of rotation
Examples
| Shape | Angle of Rotation | Order |
|---|---|---|
| Equilateral triangle | 120° | 3 |
| Square | 90° | 4 |
| Regular pentagon | 72° | 5 |
| Regular hexagon | 60° | 6 |
| Circle | Any angle | Infinite |
| Rectangle | 180° | 2 |
| Parallelogram | 180° | 2 |
| Scalene triangle | 360° | 1 |
Worked Example (ICSE 2023, 2 marks)
'Find the order of rotational symmetry of a regular octagon.'
Solution: Angle of rotation = 360°/8 = 45°. Order = 360/45 = 8.
4. Reflection Symmetry
REFLECTION is a type of symmetry where a figure is MIRRORED across a line.
Properties of Reflection
- The image is the SAME distance behind the mirror as the object is in front.
- The image is LATERALLY INVERTED (left and right are swapped).
- The line of reflection is the PERPENDICULAR BISECTOR of the segment joining a point and its image.
Examples with Letters
| Letter | Line Symmetry | Rotational Symmetry |
|---|---|---|
| A | 1 line (vertical) | Order 1 |
| H | 2 lines (vertical, horizontal) | Order 2 |
| O | Infinite (circle) | Infinite |
| X | 2 lines (diagonal) | Order 2 |
| N | 0 lines | Order 2 |
5. Symmetry in Nature and Daily Life
- Nature: Butterfly wings, snowflakes, flower petals, human face.
- Art: Rangoli patterns, mandalas, architecture.
- Design: Logos, tiles, wallpaper patterns.
Symmetry is NOT just mathematical — it appears EVERYWHERE in nature and human design.
6. Comparing Line and Rotational Symmetry
| Feature | Line Symmetry | Rotational Symmetry |
|---|---|---|
| What divides the figure | A LINE | A POINT (centre) |
| How it matches | Folding along a line | Rotating about a point |
| Example | Butterfly | Windmill |
| No. of symmetries | Number of axes | Order of rotation |
Common Mistake
'All figures with rotational symmetry also have line symmetry.' — FALSE. Example: A parallelogram has rotational symmetry (order 2) but NO line of symmetry.
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Lines of symmetry (draw and count) | 2 marks | Very High |
| Order of rotational symmetry | 2 marks | High |
| Completing symmetrical figures | 2-3 marks | Medium |
| Difference between line and rotational symmetry | 2 marks | Low |
| Symmetry in letters/numbers | 1-2 marks | Medium |
Common Mistakes in ICSE Exams
- Counting the same line twice — each distinct line counts once.
- Saying a figure has 'no symmetry' when it actually has rotational symmetry of order 1 (every figure has order 1 — it looks the same after 360°).
- Drawing lines of symmetry that are NOT valid (halves must be EXACTLY identical).
- Forgetting that a circle has INFINITE lines of symmetry, not zero.
Self-Test (5 Questions)
Q1. How many lines of symmetry does a square have? (1 mark)
- A) 2
- B) 3
- C) 4
- D) Infinite
Q2. What is the order of rotational symmetry of a rectangle? (1 mark)
Q3. Which letter has BOTH horizontal and vertical line symmetry? (1 mark)
- A) A
- B) H
- C) M
- D) T
Q4. 'A figure looks the same after a rotation of 72°. Find its order of rotational symmetry.' (2 marks)
Q5. 'Does a parallelogram have line symmetry? Does it have rotational symmetry?' (2 marks)
Answers
A1. C) 4 (two diagonals and two through midpoints of opposite sides). A2. Order 2 (looks the same after 180° and 360°). A3. B) H (vertical and horizontal lines of symmetry). A4. Order = 360/72 = 5. A5. Parallelogram has NO line symmetry. It HAS rotational symmetry of order 2.
