Symmetry
1. Introduction to Symmetry
A figure is symmetric if it can be divided or rotated so that one part matches the other exactly.
Symmetry is everywhere in nature: butterfly wings, flowers, snowflakes, and in art and architecture.
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 the line of symmetry.
Examples
| Shape | Number of Lines of Symmetry | Diagram |
|---|---|---|
| Square | 4 | 2 diagonals + 2 midlines |
| Rectangle | 2 | 2 midlines (not diagonals) |
| Equilateral triangle | 3 | 3 medians/altitudes |
| Isosceles triangle | 1 | 1 median to the base |
| Circle | Infinite | Any line through center |
| Regular hexagon | 6 | 3 through opposite vertices, 3 through midpoints |
| Scalene triangle | 0 | No line of symmetry |
Worked Example: How many lines of symmetry does a regular pentagon have?
A regular pentagon has 5 lines of symmetry (each from a vertex to the midpoint of the opposite side).
Common Mistake: Thinking a rectangle has 4 lines of symmetry (along the diagonals). Folding a rectangle along a diagonal does NOT make the halves match. A rectangle has only 2 lines of symmetry.
Exam Focus (2 marks): 'Draw a rectangle and show all its lines of symmetry.'
Draw a rectangle. Show the vertical line through the midpoints of the top and bottom sides, and the horizontal line through the midpoints of the left and right sides.
3. Rotational Symmetry
A figure has rotational symmetry if it looks the same after being rotated by an angle less than 360.
- Order of rotational symmetry: The number of times a figure fits onto itself in one full 360 rotation.
- Angle of rotation: The smallest angle through which the figure can be rotated to look the same.
| Shape | Order | Angle of Rotation |
|---|---|---|
| Equilateral triangle | 3 | 120 |
| Square | 4 | 90 |
| Rectangle | 2 | 180 |
| Regular hexagon | 6 | 60 |
| Circle | Infinite | Any angle |
| Parallelogram | 2 | 180 |
Worked Example: Find the order of rotational symmetry of a regular pentagon.
A regular pentagon has 5 equal sides and 5 equal angles. It looks the same at 72 (360/5), 144, 216, 288. Order = 5.
Common Mistake: Saying a parallelogram has no rotational symmetry. It has order 2 (looks the same after 180 rotation).
4. Reflection
Reflection is a type of symmetry where a figure is flipped across a line (mirror line).
Properties:
- The image is the same size as the object.
- The image is at the same distance from the mirror line as the object.
- Left and right are swapped in the image.
Example: Write the reflection of the word 'MOM' in a vertical mirror.
MOM is symmetric: when reflected vertically, it still reads MOM (palindromic and symmetric).
Common Mistake: Thinking a reflected image is the same as a rotated image. A reflection flips the orientation (mirror image), while a rotation turns the figure.
5. Comparing Line and Rotational Symmetry
| Aspect | Line Symmetry | Rotational Symmetry |
|---|---|---|
| Movement | Folding/reflection | Turning/rotation |
| Axis | A line | A point (center) |
| Condition | Halves match exactly | Fits onto itself during rotation |
| Measure | Number of lines | Order of rotation |
6. Symmetry in Letters
Letters with line symmetry: A, B, C, D, E (horizontal), H, I, K, M, O, T, U, V, W, X, Y.
Letters with rotational symmetry: H, I, N, O, S, X, Z (order 2 for most).
7. Self-Test
- How many lines of symmetry does a regular octagon have?
- Draw an isosceles triangle and show its line of symmetry.
- Find the order of rotational symmetry of a square.
- A figure has rotational symmetry of order 4. What is its angle of rotation?
- Which letters of the word 'MOM' are symmetrical about a vertical line?
- True or false: A circle has infinite lines of symmetry.
- Does a scalene triangle have any line of symmetry? Explain.
8. Answers to Self-Test
- 8 lines of symmetry.
- Draw a triangle with two equal sides. Draw the line from the vertex between the equal sides to the midpoint of the base.
- Order 4 (angles: 90, 180, 270, 360).
- 360/4 = 90.
- M (vertical symmetry), O (vertical symmetry). M is symmetric about a vertical line; O is symmetric about both vertical and horizontal lines.
- True. Any line through the center is a line of symmetry.
- How many lines of symmetry does a regular hexagon have? (Answer: 6 — 3 through opposite vertices and 3 through midpoints of opposite sides.)
Symmetry in Art and Architecture
Symmetry is fundamental to beauty and design. The Taj Mahal is perfectly symmetrical — the left half mirrors the right half exactly. Rangoli designs use rotational symmetry. 'Indian architecture — from temples to mosques to palaces — is a masterclass in symmetry. The Lepakshi Temple in Andhra Pradesh and the Sun Temple at Konark both demonstrate extraordinary geometric symmetry in their design. Learning symmetry in math helps you appreciate the beauty in the world around you.'
ICSE Exam Tips
'In ICSE Class 6, symmetry questions are visual and fun but carry 3-4 marks. When asked to complete a symmetric figure, use a MIRROR to check your work. For rotational symmetry, trace the shape on tracing paper, pin the centre, and rotate to verify. Always state both line symmetry AND rotational symmetry where applicable.' 7. No. A scalene triangle has no equal sides, so no fold line can make the two halves match.
