Symmetry (Grade 5)
Symmetry means that one part of a figure is the mirror image of the other part. When you fold a symmetric figure along a special line, both halves match exactly.
Symmetry is one of the most beautiful ideas in mathematics. It appears everywhere in nature — in butterflies, flowers, snowflakes, and even the human face. Artists, architects, and designers use symmetry to create balanced and attractive patterns.
In Class 5, you will learn about line symmetry (reflection symmetry) and get an introduction to rotational symmetry. You will find lines of symmetry in shapes, letters, and everyday objects, and determine the order of rotational symmetry for common figures.
What is Symmetry - Class 5 Maths (Geometry)?
A figure has line symmetry if there is a line (called the line of symmetry or mirror line) that divides the figure into two identical halves. When you fold the figure along this line, both halves overlap perfectly.
A figure has rotational symmetry if it looks the same after being rotated (turned) by less than a full turn (360°) about its centre. The number of times a figure matches itself during one full turn is called its order of rotational symmetry.
Types and Properties
Line Symmetry:
- A figure can have 0, 1, 2, or more lines of symmetry.
- The line of symmetry can be vertical, horizontal, or diagonal.
Lines of symmetry in common shapes:
| Shape | Lines of Symmetry |
|---|---|
| Circle | Infinite |
| Square | 4 |
| Rectangle | 2 |
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
- A square has rotational symmetry of order 4 (matches at 90°, 180°, 270°, 360°).
- An equilateral triangle has rotational symmetry of order 3 (matches at 120°, 240°, 360°).
- A rectangle has rotational symmetry of order 2 (matches at 180° and 360°).
Solved Examples
Example 1: Example 1: Lines of Symmetry in a Square
Problem: How many lines of symmetry does a square have? Describe them.
Solution:
Step 1: A square has 4 lines of symmetry:
- 1 vertical line (top to bottom through the middle)
- 1 horizontal line (left to right through the middle)
- 2 diagonal lines (corner to opposite corner)
Answer: A square has 4 lines of symmetry.
Example 2: Example 2: Line Symmetry in Letters
Problem: Which capital letters have a vertical line of symmetry: A, B, C, M, S?
Solution:
Step 1: Check each letter by imagining a vertical fold:
- A — left and right halves match → Yes
- B — left and right halves do NOT match → No
- C — left and right halves do NOT match → No
- M — left and right halves match → Yes
- S — left and right halves do NOT match → No
Answer: A and M have a vertical line of symmetry.
Example 3: Example 3: Rectangle Symmetry
Problem: Does a rectangle have a diagonal line of symmetry?
Solution:
Step 1: Draw a rectangle and fold it along its diagonal.
Step 2: The two halves do NOT overlap perfectly (since the length and breadth are different).
Step 3: A rectangle has only 2 lines of symmetry: one vertical and one horizontal (through the midpoints of opposite sides).
Answer: No, a rectangle does not have a diagonal line of symmetry.
Example 4: Example 4: No Line of Symmetry
Problem: Does a scalene triangle have any line of symmetry?
Solution:
Step 1: A scalene triangle has all sides and all angles different.
Step 2: There is no way to fold it so that both halves match.
Answer: No, a scalene triangle has 0 lines of symmetry.
Example 5: Example 5: Symmetry in Nature
Problem: A butterfly has one line of symmetry. Where is it?
Solution:
Step 1: A butterfly’s left wing is the mirror image of the right wing.
Step 2: The line of symmetry runs down the body of the butterfly from head to tail (a vertical line).
Answer: The line of symmetry is along the body of the butterfly (vertical line).
Example 6: Example 6: Drawing a Symmetric Figure
Problem: Half of a symmetric figure is given on the left side of a vertical mirror line. Complete the figure.
Solution:
Step 1: For each point on the left half, mark a point at the same distance on the right side of the mirror line.
Step 2: Connect the points in the same order to complete the right half.
Step 3: The completed figure should be a mirror image of the left half.
Answer: The figure is completed by reflecting each point across the mirror line.
Example 7: Example 7: Rotational Symmetry of a Square
Problem: What is the order of rotational symmetry of a square?
Solution:
Step 1: Rotate a square about its centre. It looks the same at:
- 90° (quarter turn)
- 180° (half turn)
- 270° (three-quarter turn)
- 360° (full turn)
Step 2: The square matches itself 4 times in one full rotation.
Answer: The order of rotational symmetry is 4.
Example 8: Example 8: Equilateral Triangle Symmetry
Problem: How many lines of symmetry and what order of rotational symmetry does an equilateral triangle have?
Solution:
Step 1: Lines of symmetry: 3 (one from each vertex to the midpoint of the opposite side).
Step 2: Rotational symmetry: matches at 120°, 240°, 360° → order = 3.
Answer: 3 lines of symmetry and rotational symmetry of order 3.
Example 9: Example 9: Regular Hexagon
Problem: How many lines of symmetry does a regular hexagon have?
Solution:
Step 1: A regular hexagon has 6 equal sides.
Step 2: It has 3 lines connecting opposite vertices and 3 lines connecting midpoints of opposite sides.
Answer: A regular hexagon has 6 lines of symmetry.
Example 10: Example 10: Identifying Symmetry in Rangoli
Problem: Meera draws a rangoli pattern that looks the same when rotated 90°. What is the order of rotational symmetry?
Solution:
Step 1: If the pattern repeats every 90°, it matches at 90°, 180°, 270°, and 360°.
Step 2: That is 4 matches in a full turn.
Answer: The order of rotational symmetry is 4.
Real-World Applications
Where do we see symmetry?
- Nature: Butterflies have bilateral symmetry (1 line). Starfish have 5-fold rotational symmetry. Snowflakes have 6-fold symmetry. Flowers often have 4, 5, or 6 lines of symmetry.
- Architecture: The Taj Mahal is one of the most famous symmetric buildings in the world. Temples, mosques, churches, and government buildings use symmetry for balance and grandeur.
- Art: Rangoli patterns (Diwali), mehndi designs, kolam, and Islamic geometric patterns all use symmetry extensively.
- Everyday objects: Human faces (approximately symmetric), spectacles, shirts when buttoned, and scissors.
- Flags: The Indian national flag has a horizontal line of symmetry (if we ignore the Ashoka Chakra details). Many flags have vertical symmetry.
- Letters and numbers: Letters like A, H, M, O, T, U, V, W, X, Y have at least one line of symmetry. The number 8 has two lines of symmetry.
Key Points to Remember
- A figure has line symmetry if a line divides it into two identical mirror halves.
- The line that divides the figure is called the line of symmetry (or mirror line).
- A figure can have 0, 1, 2, or many lines of symmetry.
- A circle has infinite lines of symmetry. A square has 4. A rectangle has 2.
- Rotational symmetry means the figure looks the same after a partial turn.
- The order of rotational symmetry is the number of times a figure matches itself in one full turn.
- A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
Practice Problems
- How many lines of symmetry does an isosceles triangle have?
- List three capital English letters that have exactly one line of symmetry.
- A regular pentagon has how many lines of symmetry?
- Draw the other half of a shape given its left half and a vertical line of symmetry.
- Does the letter Z have a line of symmetry? Does it have rotational symmetry?
- What is the order of rotational symmetry of a rectangle?
- Name two objects in your classroom that have line symmetry.
- Can a figure have rotational symmetry but no line symmetry? Give an example.
Frequently Asked Questions
Q1. What is line symmetry?
Line symmetry (or reflection symmetry) means a figure can be divided by a line into two halves that are exact mirror images of each other. When folded along the line, both halves overlap completely.
Q2. What is a line of symmetry?
A line of symmetry is the line along which you fold a figure so that both halves match exactly. It is also called a mirror line.
Q3. How many lines of symmetry does a circle have?
A circle has infinitely many lines of symmetry. Any diameter of the circle is a line of symmetry.
Q4. What is rotational symmetry?
A figure has rotational symmetry if it looks the same after being rotated by an angle less than 360° about its centre. The order of rotational symmetry is how many times it matches itself in one full rotation.
Q5. Does a rectangle have diagonal lines of symmetry?
No. A rectangle (that is not a square) has only 2 lines of symmetry: one vertical and one horizontal, both passing through the centre. The diagonals are not lines of symmetry.
Q6. Can a figure have no lines of symmetry?
Yes. A scalene triangle and a parallelogram (that is not a rectangle or rhombus) have no lines of symmetry.
Q7. What is the difference between line symmetry and rotational symmetry?
Line symmetry involves folding along a line to match two halves. Rotational symmetry involves spinning the figure around a centre point. A figure can have one type, both types, or neither.
Q8. How do I find lines of symmetry in a shape?
Try folding the shape (or imagining a fold) along different lines — vertical, horizontal, and diagonal. If both halves match exactly on a fold, that fold line is a line of symmetry.
Q9. Does the letter S have symmetry?
The letter S has no line of symmetry but has rotational symmetry of order 2 — it looks the same when rotated 180°.
Q10. Is symmetry covered in the NCERT Class 5 syllabus?
Yes. Line symmetry and an introduction to rotational symmetry are part of the Geometry chapter in NCERT/CBSE Class 5 Maths.
Related Topics
- Symmetry (Grade 4)
- Reflection and Rotation
- Lines, Line Segments and Rays
- Parallel and Perpendicular Lines
- Angles (Grade 5)
- Angle Sum Property of Triangle
- Types of Triangles (Grade 5)
- Quadrilaterals (Grade 5)
- Circles (Grade 5)
- Nets of 3D Shapes (Grade 5)
- Views of 3D Shapes (Grade 5)
- Complementary and Supplementary Angles
