Line of Symmetry
Fold a piece of paper in half. Cut a shape along the fold. When you open it, both halves look exactly the same. The fold line is called a line of symmetry.
Symmetry means that one half of a figure is the mirror image of the other half. When you fold a symmetric shape along its line of symmetry, the two halves match perfectly.
Symmetry is found everywhere — in butterflies, leaves, buildings, rangoli patterns, and even the letters of the alphabet. Recognising symmetry helps us understand shapes better and is used in art, design, and architecture.
In this chapter, you will learn what a line of symmetry is, how to find lines of symmetry in different shapes, and how many lines of symmetry common figures have.
What is Line of Symmetry - Grade 6 Maths (Symmetry)?
Definition: A line of symmetry is a line that divides a figure into two parts such that one part is the exact mirror image of the other. If you fold the figure along this line, the two halves overlap perfectly.
A line of symmetry is also called an axis of symmetry or a mirror line.
Key points:
- A shape can have 0, 1, 2, or more lines of symmetry.
- If a shape has no line of symmetry, it is called asymmetric.
- The line of symmetry can be vertical, horizontal, or diagonal.
- When a shape is folded along a line of symmetry, both halves must match exactly — same size and same shape.
Symmetry test: Place a mirror along the line. If the reflection combined with the visible half looks exactly like the original shape, then the line is a line of symmetry.
Line of Symmetry Formula
Lines of symmetry in common shapes:
| Shape | Lines of Symmetry |
| Scalene triangle | 0 |
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Rectangle | 2 |
| Square | 4 |
| Rhombus | 2 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | Infinite (unlimited) |
Rule for regular polygons:
A regular polygon with n sides has exactly n lines of symmetry.
Derivation and Proof
How to check if a line is a line of symmetry:
Method 1: Paper Folding
- Draw or cut out the shape on paper.
- Try folding along a line.
- If both halves match exactly when folded, it is a line of symmetry.
- Try different folds (vertical, horizontal, diagonal) to find all lines.
Method 2: Mirror Test
- Place a small mirror along the line you want to test.
- Look at the reflection in the mirror combined with the visible half.
- If the result looks exactly like the original shape, the line is a line of symmetry.
Example — finding lines of symmetry of a rectangle:
- Fold vertically (along the centre from top to bottom): Left half matches right half. This is a line of symmetry.
- Fold horizontally (along the centre from left to right): Top half matches bottom half. This is a line of symmetry.
- Fold diagonally: The halves do NOT match (corners stick out). This is NOT a line of symmetry.
So a rectangle has 2 lines of symmetry (vertical and horizontal).
Why a square has 4 lines of symmetry:
- 1 vertical line (through the middle)
- 1 horizontal line (through the middle)
- 2 diagonal lines (corner to corner)
- Total: 4 lines of symmetry
Types and Properties
Types of symmetry problems:
Type 1: Identify lines of symmetry
- Given a shape, find how many lines of symmetry it has and draw them.
Type 2: Complete a symmetric figure
- Given half a shape and the line of symmetry, draw the other half.
Type 3: Symmetry in letters and numbers
- Identify which English letters have vertical, horizontal, or both lines of symmetry.
- Example: A, M, T, U, V, W, Y have a vertical line of symmetry.
Type 4: Symmetry in nature and everyday objects
- Identify symmetry in butterflies, leaves, faces, buildings, etc.
Type 5: Regular polygons and symmetry
- Use the rule: a regular polygon with n sides has n lines of symmetry.
Type 6: True or False questions
- "A rectangle has 4 lines of symmetry" — False (it has 2).
- "A circle has infinite lines of symmetry" — True.
Solved Examples
Example 1: Example 1: Lines of Symmetry of a Square
Problem: How many lines of symmetry does a square have? Describe each one.
Solution:
- Line 1: Vertical line through the middle (left-right fold).
- Line 2: Horizontal line through the middle (top-bottom fold).
- Line 3: Diagonal line from top-left to bottom-right.
- Line 4: Diagonal line from top-right to bottom-left.
Answer: A square has 4 lines of symmetry.
Example 2: Example 2: Lines of Symmetry of a Rectangle
Problem: How many lines of symmetry does a rectangle (that is not a square) have?
Solution:
- Vertical fold: left half matches right half — Yes, this is a line of symmetry.
- Horizontal fold: top half matches bottom half — Yes, this is a line of symmetry.
- Diagonal fold: halves do NOT match — No, not a line of symmetry.
Answer: A rectangle has 2 lines of symmetry (vertical and horizontal only).
Example 3: Example 3: Equilateral Triangle
Problem: Find the lines of symmetry of an equilateral triangle.
Solution:
- An equilateral triangle has 3 equal sides.
- Each line of symmetry goes from one vertex to the midpoint of the opposite side.
- There are 3 such lines.
Answer: An equilateral triangle has 3 lines of symmetry.
Example 4: Example 4: Symmetry in English Letters
Problem: Which of these letters have a line of symmetry: A, B, C, F, M, S, X?
Solution:
- A — vertical line of symmetry (left-right mirror).
- B — horizontal line of symmetry (top-bottom mirror).
- C — horizontal line of symmetry.
- F — no line of symmetry.
- M — vertical line of symmetry.
- S — no line of symmetry.
- X — both vertical and horizontal lines of symmetry.
Answer: A, B, C, M, and X have lines of symmetry. F and S do not.
Example 5: Example 5: Regular Hexagon
Problem: How many lines of symmetry does a regular hexagon have?
Solution:
- A regular hexagon has 6 equal sides.
- Rule: A regular polygon with n sides has n lines of symmetry.
- n = 6, so lines of symmetry = 6.
Answer: A regular hexagon has 6 lines of symmetry.
Example 6: Example 6: Isosceles Triangle
Problem: An isosceles triangle has two equal sides. How many lines of symmetry does it have?
Solution:
- The line from the vertex angle (where the two equal sides meet) to the midpoint of the base is a line of symmetry.
- No other fold creates matching halves.
Answer: An isosceles triangle has 1 line of symmetry.
Example 7: Example 7: Circle
Problem: How many lines of symmetry does a circle have?
Solution:
- Any line passing through the centre of a circle divides it into two equal halves.
- You can draw such a line at any angle — there are infinitely many.
Answer: A circle has infinite (unlimited) lines of symmetry.
Example 8: Example 8: Completing a Symmetric Figure
Problem: The left half of a symmetric shape shows a triangle. The line of symmetry is vertical. Describe what the complete shape looks like.
Solution:
- Draw the mirror image of the triangle on the right side of the vertical line.
- Each point on the left half has a matching point at the same distance on the right side of the line.
- The complete shape will be a symmetric figure — like a diamond or kite shape depending on the triangle.
Example 9: Example 9: Scalene Triangle
Problem: Does a scalene triangle have any line of symmetry?
Solution:
- A scalene triangle has all three sides of different lengths.
- No fold can create two matching halves.
Answer: A scalene triangle has 0 lines of symmetry.
Example 10: Example 10: Rhombus
Problem: How many lines of symmetry does a rhombus have?
Solution:
- A rhombus has 4 equal sides, but its angles are not 90 degrees (unless it is a square).
- Fold along the longer diagonal: both halves match — line of symmetry.
- Fold along the shorter diagonal: both halves match — line of symmetry.
- Fold vertically or horizontally through midpoints of sides: halves do NOT match.
Answer: A rhombus has 2 lines of symmetry (both diagonals).
Real-World Applications
Symmetry in real life:
- Nature: Butterflies, flowers, starfish, and snowflakes all show symmetry. A butterfly has one line of symmetry (down its body).
- Architecture: The Taj Mahal, many temples, and modern buildings are designed with symmetry for beauty and balance.
- Art and design: Rangoli patterns, kolam designs, and mandalas use multiple lines of symmetry.
- Human body: The human face has approximate vertical symmetry — the left side roughly mirrors the right side.
- Logos: Many company logos use symmetry to look balanced and memorable.
- Alphabets: Many English capital letters have symmetry, which helps in understanding mirror writing and reflections.
Key Points to Remember
- A line of symmetry divides a figure into two halves that are mirror images of each other.
- If you fold a shape along its line of symmetry, both halves overlap exactly.
- A shape can have 0, 1, 2, or more lines of symmetry.
- A regular polygon with n sides has exactly n lines of symmetry.
- A circle has infinite lines of symmetry (any diameter is a line of symmetry).
- A square has 4 lines of symmetry. A rectangle has 2. An equilateral triangle has 3.
- A scalene triangle has 0 lines of symmetry.
- The line of symmetry can be vertical, horizontal, or diagonal.
- Symmetry is found in nature, art, architecture, and design.
- The mirror test and paper folding are two methods to check for lines of symmetry.
Practice Problems
- How many lines of symmetry does a regular pentagon have?
- Which of these shapes has exactly 1 line of symmetry: square, isosceles triangle, circle, scalene triangle?
- List all English capital letters that have exactly one vertical line of symmetry.
- Does a parallelogram (that is not a rectangle) have any line of symmetry?
- Draw a shape that has exactly 2 lines of symmetry.
- An arrow pointing right — does it have a line of symmetry? If yes, is it vertical or horizontal?
- How many lines of symmetry does a regular octagon (8 sides) have?
- Complete this symmetric figure: the left half shows a right-angled triangle with the right angle at the bottom-left. The line of symmetry is vertical.
Frequently Asked Questions
Q1. What is a line of symmetry?
A line of symmetry is a line that divides a figure into two halves that are mirror images of each other. If you fold the shape along this line, the two halves overlap perfectly.
Q2. Can a shape have no line of symmetry?
Yes. A scalene triangle, a parallelogram (that is not a rectangle), and many irregular shapes have no line of symmetry. These shapes are called asymmetric.
Q3. How do I find lines of symmetry?
Use the paper folding method: fold the shape along a line and check if both halves match. Try vertical, horizontal, and diagonal folds. Each fold that creates matching halves is a line of symmetry.
Q4. Why does a rectangle have only 2 lines of symmetry (not 4)?
A rectangle has 2 lines: one vertical and one horizontal through the centre. The diagonals are NOT lines of symmetry because when you fold along a diagonal, the corners do not overlap (the rectangle is longer than it is wide).
Q5. How many lines of symmetry does a circle have?
A circle has infinite (unlimited) lines of symmetry. Any line through the centre (diameter) divides the circle into two equal halves. Since you can draw a diameter at any angle, there are infinitely many.
Q6. Is the diagonal of a rectangle a line of symmetry?
No (unless the rectangle is a square). When you fold a rectangle along its diagonal, the two halves do not overlap because the length and breadth are different.
Q7. What is the rule for regular polygons?
A regular polygon with n sides has exactly n lines of symmetry. An equilateral triangle (3 sides) has 3 lines, a square (4 sides) has 4 lines, a regular pentagon (5 sides) has 5 lines, and so on.
Q8. Does the letter S have a line of symmetry?
No. The letter S does not have a line of symmetry. If you fold it vertically or horizontally, the two halves do not match. S does have rotational symmetry (it looks the same when rotated 180 degrees), but that is different from line symmetry.
Q9. What is the difference between line symmetry and rotational symmetry?
Line symmetry means a shape can be divided into two matching halves by a line (fold and match). Rotational symmetry means a shape looks the same after being rotated by some angle less than 360 degrees. A shape can have one, both, or neither.
Q10. Can a shape have exactly 3 lines of symmetry?
Yes. An equilateral triangle has exactly 3 lines of symmetry. Each line goes from a vertex to the midpoint of the opposite side.
