Notes For All Chapters – Maths Class 6 Ganita Prakash
📚 Class 6 • Mathematics • Ganita Prakash
✦ Chapter 9: Symmetry ✦
Explore the beautiful world of symmetry — from butterflies and rangoli to squares and circles!
🪞 Line of Symmetry
🔄 Rotational Symmetry
📐 Reflection
⭕ Circle Symmetry
📝 Exam Questions
🔄 Rotational Symmetry
📐 Reflection
⭕ Circle Symmetry
📝 Exam Questions
📋 Contents
Introduction to Symmetry
Look around you — a butterfly, a rangoli, a pinwheel, or even the Taj Mahal. There is something special about these objects. Their beauty comes from a hidden pattern called symmetry.
Why does a butterfly look beautiful?
Both wings of a butterfly are identical mirror images of each other. The left wing and the right wing are exactly the same — this is symmetry in nature!
Both wings of a butterfly are identical mirror images of each other. The left wing and the right wing are exactly the same — this is symmetry in nature!
🔍 What is Symmetry?
A figure has symmetry when it is made up of parts that repeat in a definite pattern.
Such a figure is called symmetrical.
Such a figure is called symmetrical.
A cloud, on the other hand, has no such repeating pattern — it is asymmetrical (not symmetrical).
✅ Symmetrical Examples
Butterfly, Flower, Rangoli, Pinwheel, Square, Equilateral Triangle, Circle, Taj Mahal
Butterfly, Flower, Rangoli, Pinwheel, Square, Equilateral Triangle, Circle, Taj Mahal
❌ Asymmetrical Examples
Clouds, Amoeba, Irregular shapes, Most natural rocks and stones
Clouds, Amoeba, Irregular shapes, Most natural rocks and stones
🏛️
Symmetry in Architecture!
The Taj Mahal has perfect reflection symmetry — its left half is a mirror image of its right half. The Gopuram (temple tower) also shows beautiful symmetry patterns built by ancient Indian architects.
The Taj Mahal has perfect reflection symmetry — its left half is a mirror image of its right half. The Gopuram (temple tower) also shows beautiful symmetry patterns built by ancient Indian architects.
9.1 Line of Symmetry
Take a piece of paper shaped like a triangle. Fold it along a dotted line through the middle. If one half covers the other half completely — those two parts are called mirror halves!
A line of symmetry (also called axis of symmetry) is a line that cuts a figure into two parts that exactly overlap when folded along that line.
Left: Triangle with 1 line of symmetry | Right: Shape with no line of symmetry
Key Test for Line of Symmetry:
Fold the shape along the line. If both halves match exactly (one sits perfectly on top of the other), then that line IS a line of symmetry. If they don’t match, it is NOT a line of symmetry.
Fold the shape along the line. If both halves match exactly (one sits perfectly on top of the other), then that line IS a line of symmetry. If they don’t match, it is NOT a line of symmetry.
📐 Lines of Symmetry in Common Shapes
| Shape | Number of Lines of Symmetry | Where are the Lines? |
|---|---|---|
| Line Segment | 2 | Perpendicular bisector + the line itself |
| Equilateral Triangle | 3 | From each vertex to midpoint of opposite side |
| Square | 4 | 2 through midpoints of sides + 2 diagonals |
| Rectangle (non-square) | 2 | Through midpoints of opposite sides only |
| Isosceles Triangle | 1 | From apex to base midpoint |
| Scalene Triangle | 0 | None |
| Regular Pentagon | 5 | From each vertex to midpoint of opposite side |
| Regular Hexagon | 6 | 3 through vertices + 3 through midpoints |
| Circle | Infinite | Every diameter is a line of symmetry! |
Common Mistake:
A rectangle’s diagonal is NOT a line of symmetry (the two parts don’t overlap when you fold). But a square’s diagonal IS a line of symmetry. Don’t confuse these two!
A rectangle’s diagonal is NOT a line of symmetry (the two parts don’t overlap when you fold). But a square’s diagonal IS a line of symmetry. Don’t confuse these two!
Figures with Multiple Lines of Symmetry
Some figures have more than one line of symmetry. The square is a great example. Let’s discover all its lines of symmetry by paper folding!
🗂️ Folding a Square – Finding All 4 Lines
- Fold 1 (Vertical): Fold the square in half from left to right. Open it. You get the vertical line of symmetry.
- Fold 2 (Horizontal): Fold the square in half from top to bottom. Open it. You get the horizontal line of symmetry.
- Fold 3 (Diagonal 1): Fold along the diagonal from top-left to bottom-right. You get the first diagonal line of symmetry.
- Fold 4 (Diagonal 2): Fold along the other diagonal (top-right to bottom-left). You get the second diagonal line of symmetry.
A square has 4 lines of symmetry: vertical, horizontal, and 2 diagonals
A regular polygon with n sides has exactly n lines of symmetry.
Examples: Triangle (3 sides → 3 lines), Square (4 sides → 4 lines), Pentagon (5 sides → 5 lines)
Examples: Triangle (3 sides → 3 lines), Square (4 sides → 4 lines), Pentagon (5 sides → 5 lines)
Quick Tip:
For regular polygons, draw lines from each vertex to the midpoint of the opposite side (or opposite vertex for even-sided polygons). That gives you all the lines of symmetry!
For regular polygons, draw lines from each vertex to the midpoint of the opposite side (or opposite vertex for even-sided polygons). That gives you all the lines of symmetry!
Reflection Symmetry
When we fold a figure along its line of symmetry, one part gets reflected onto the other. This is called Reflection Symmetry.
A figure that has a line or lines of symmetry is said to have Reflection Symmetry.
The line of symmetry acts like a mirror — reflecting one half onto the other.
The line of symmetry acts like a mirror — reflecting one half onto the other.
🔠 Reflection of Points in a Square
Label the corners of a square as A (top-left), B (top-right), C (bottom-right), D (bottom-left).
Reflecting a square along the vertical line of symmetry
Reflection Rule:
When you reflect a figure along a line of symmetry, every point on one side moves to a corresponding position on the other side. The line of symmetry is equidistant (equal distance) from both corresponding points.
When you reflect a figure along a line of symmetry, every point on one side moves to a corresponding position on the other side. The line of symmetry is equidistant (equal distance) from both corresponding points.
✍️ Letters with Reflection Symmetry
Vertical Line of Symmetry
Letters: A, H, I, M, O, T, U, V, W, X, Y
(Left half = Right half)
Letters: A, H, I, M, O, T, U, V, W, X, Y
(Left half = Right half)
Horizontal Line of Symmetry
Letters: B, C, D, E, H, I, K, O, X
(Top half = Bottom half)
Letters: B, C, D, E, H, I, K, O, X
(Top half = Bottom half)
Generating Symmetric Shapes
We can create symmetrical figures using fun activities like ink blots, paper folding, and paper cutting!
🖊️ Activity 1: Ink Blot Devils
- Take a piece of paper and fold it in half.
- Open the paper. Put a few drops of ink (or paint) on one half.
- Press both halves together firmly.
- Open the paper again and see the symmetric pattern created!
- The fold line is the line of symmetry of the ink blot figure.
Why is the ink blot symmetric?
Because the wet ink gets pressed onto both halves equally, creating mirror images on each side of the fold. The fold is the line of symmetry!
Because the wet ink gets pressed onto both halves equally, creating mirror images on each side of the fold. The fold is the line of symmetry!
✂️ Activity 2: Paper Folding and Cutting
- Take a square piece of paper.
- Fold it in half (vertical or horizontal fold).
- Cut out a shape from the folded edge or from the open side.
- Unfold the paper. The resulting shape will be symmetric!
- The fold line becomes the line of symmetry.
Key Insight:
Whenever you fold paper and cut, the fold line automatically becomes a line of symmetry. If you fold twice before cutting, you can get figures with 2 lines of symmetry. Multiple folds = multiple lines of symmetry!
Whenever you fold paper and cut, the fold line automatically becomes a line of symmetry. If you fold twice before cutting, you can get figures with 2 lines of symmetry. Multiple folds = multiple lines of symmetry!
🎯 Activity 3: Punching Game
Fold a square sheet of paper. Punch a hole through it. When you unfold, the two holes are symmetric about the fold line.
1 Fold → 2 holes
The fold line is the line of symmetry. Both holes are mirror images of each other.
The fold line is the line of symmetry. Both holes are mirror images of each other.
2 Folds → 4 holes
Two fold lines → two lines of symmetry. All 4 holes are symmetric about both fold lines.
Two fold lines → two lines of symmetry. All 4 holes are symmetric about both fold lines.
9.2 Rotational Symmetry
A paper windmill looks symmetrical, but it has no line of symmetry. However, if you rotate it by 90° around its centre, it looks exactly the same! This is a different kind of symmetry called Rotational Symmetry.
A figure has Rotational Symmetry if it looks exactly the same after being rotated by some angle (strictly between 0° and 360°) about a fixed point.
The fixed point about which the rotation happens is called the Centre of Rotation.
📐 Key Definitions
🔵 Angle of Symmetry
The angle by which a figure can be rotated to look exactly the same as the original. Also called “angle of rotational symmetry”.
The angle by which a figure can be rotated to look exactly the same as the original. Also called “angle of rotational symmetry”.
🔵 Centre of Rotation
The fixed point about which the figure is rotated. It is usually the centre of the figure.
The fixed point about which the figure is rotated. It is usually the centre of the figure.
A windmill looks the same after every 90° rotation — it has 4 angles of symmetry
360° is ALWAYS an angle of symmetry!
Every figure looks the same after a complete rotation of 360° (full turn), because it returns to its original position. So 360° is always an angle of symmetry for any figure.
Every figure looks the same after a complete rotation of 360° (full turn), because it returns to its original position. So 360° is always an angle of symmetry for any figure.
Important:
A figure is said to have rotational symmetry only if there is an angle of symmetry strictly between 0° and 360°. A figure that only “comes back” at 360° does NOT have rotational symmetry.
A figure is said to have rotational symmetry only if there is an angle of symmetry strictly between 0° and 360°. A figure that only “comes back” at 360° does NOT have rotational symmetry.
Angles of Symmetry — Finding Them
🔢 The Formula for Angles of Symmetry
For a figure with n angles of symmetry (i.e., order of rotational symmetry = n), the smallest angle is:
Smallest Angle of Symmetry = 360° ÷ n
And the other angles are: smallest angle × 2, × 3, × 4, …, up to 360°
📊 Angles of Symmetry for Common Cases
| No. of Angles (Order) | Smallest Angle | All Angles of Symmetry | Example |
|---|---|---|---|
| 2 | 180° | 180°, 360° | Rectangle, Letter S, Letter Z |
| 3 | 120° | 120°, 240°, 360° | Equilateral Triangle |
| 4 | 90° | 90°, 180°, 270°, 360° | Square, Windmill (4 blades) |
| 5 | 72° | 72°, 144°, 216°, 288°, 360° | Regular Pentagon, Star |
| 6 | 60° | 60°, 120°, 180°, 240°, 300°, 360° | Regular Hexagon, Snowflake |
| ∞ | Any angle | Every angle! | Circle |
Pattern Observed:
In each set, all the angles of symmetry are multiples of the smallest angle. For example, for a square (smallest = 90°), the angles are 90°, 180° (=90°×2), 270° (=90°×3), 360° (=90°×4).
In each set, all the angles of symmetry are multiples of the smallest angle. For example, for a square (smallest = 90°), the angles are 90°, 180° (=90°×2), 270° (=90°×3), 360° (=90°×4).
🔑 Key Rule: Factors of 360°
If the smallest angle of symmetry is a whole number in degrees, then it is always a factor of 360.
Factors of 360 include: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
Factors of 360 include: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
📏 How to Find Angles for Radial Arm Figures
For a figure with n equal radial arms (like a fan), the angle between adjacent arms must be 360° ÷ n for the figure to have rotational symmetry.
2 arms → angle between arms = 360° ÷ 2 = 180°
3 arms → angle between arms = 360° ÷ 3 = 120°
4 arms → angle between arms = 360° ÷ 4 = 90°
5 arms → angle between arms = 360° ÷ 5 = 72°
6 arms → angle between arms = 360° ÷ 6 = 60°
3 arms → angle between arms = 360° ÷ 3 = 120°
4 arms → angle between arms = 360° ÷ 4 = 90°
5 arms → angle between arms = 360° ÷ 5 = 72°
6 arms → angle between arms = 360° ÷ 6 = 60°
For 7 arms:
360° ÷ 7 = 51.43°… which is not a whole number. Express it as a mixed fraction: 51 3/7 °. So 360° doesn’t always divide evenly!
360° ÷ 7 = 51.43°… which is not a whole number. Express it as a mixed fraction: 51 3/7 °. So 360° doesn’t always divide evenly!
🇮🇳
Ashoka Chakra — 24 Angles of Symmetry!
The Ashoka Chakra on the Indian national flag has 24 spokes equally spaced. Its smallest angle of symmetry = 360° ÷ 24 = 15°. It also has 24 lines of symmetry — one through each spoke!
The Ashoka Chakra on the Indian national flag has 24 spokes equally spaced. Its smallest angle of symmetry = 360° ÷ 24 = 15°. It also has 24 lines of symmetry — one through each spoke!
Symmetries of a Circle
The circle is the most symmetrical shape of all! Let’s explore its amazing symmetry properties.
🔄 Rotational Symmetry
A circle can be rotated by any angle about its centre and still looks the same. Every angle (1°, 2°, 5°, 45°, 90°, 137°…) is an angle of symmetry!
A circle can be rotated by any angle about its centre and still looks the same. Every angle (1°, 2°, 5°, 45°, 90°, 137°…) is an angle of symmetry!
🪞 Reflection Symmetry
Every diameter of a circle is a line of symmetry. Since there are infinitely many diameters, a circle has infinite lines of symmetry!
Every diameter of a circle is a line of symmetry. Since there are infinitely many diameters, a circle has infinite lines of symmetry!
A circle has infinite lines of symmetry — every diameter is one!
For a Circle:
• Lines of Symmetry = Infinite (every diameter)
• Angles of Symmetry = Every angle from 0° to 360°
• Centre of Rotation = Centre of the circle
• Lines of Symmetry = Infinite (every diameter)
• Angles of Symmetry = Every angle from 0° to 360°
• Centre of Rotation = Centre of the circle
🌀 Real-life Examples of Rotational Symmetry
- Fan (3 blades): Angles of symmetry = 120°, 240°, 360°
- Fan (4 blades): Angles of symmetry = 90°, 180°, 270°, 360°
- Car Wheel: Depends on number of spokes — if 8 spokes, smallest angle = 45°
- Sunflower: Has many angles of symmetry depending on petal count
- Snowflake: 6 angles of symmetry — 60°, 120°, 180°, 240°, 300°, 360°
Reflection vs Rotational Symmetry — Comparison
| Feature | Reflection Symmetry | Rotational Symmetry |
|---|---|---|
| Also known as | Line Symmetry / Bilateral Symmetry | Rotational / Turn Symmetry |
| Key element | Line of Symmetry (mirror line) | Centre of Rotation |
| Action | Folding/reflecting along a line | Rotating about a point |
| Result | Two halves overlap perfectly | Figure looks same after rotation |
| Example | Butterfly, Isosceles Triangle | Windmill, Equilateral Triangle |
| Can figure have both? | Yes! e.g. Square, Equilateral Triangle, Circle have BOTH | |
| Can figure have one but not other? | Yes! Isosceles triangle has reflection but NOT rotational. Windmill (4 blades, asymmetric design) can have rotation but not reflection. | |
Chapter Summary
📌 Symmetry
A figure is symmetrical if it is made up of parts that repeat in a definite pattern.
A figure is symmetrical if it is made up of parts that repeat in a definite pattern.
🪞 Line of Symmetry
A line that divides a figure into two parts that exactly overlap when folded. Also called the axis of symmetry.
A line that divides a figure into two parts that exactly overlap when folded. Also called the axis of symmetry.
✦ Multiple Lines
A figure can have more than one line of symmetry. A regular n-gon has n lines of symmetry.
A figure can have more than one line of symmetry. A regular n-gon has n lines of symmetry.
🔄 Rotational Symmetry
A figure has rotational symmetry if it looks the same after rotating by an angle strictly between 0° and 360°.
A figure has rotational symmetry if it looks the same after rotating by an angle strictly between 0° and 360°.
📐 Angle of Symmetry
The angle of rotation that makes a figure look the same. All angles are multiples of the smallest. The smallest angle is a factor of 360°.
The angle of rotation that makes a figure look the same. All angles are multiples of the smallest. The smallest angle is a factor of 360°.
⭕ Circle’s Symmetry
A circle has infinite lines of symmetry (every diameter) and every angle is an angle of rotational symmetry.
A circle has infinite lines of symmetry (every diameter) and every angle is an angle of rotational symmetry.
True or False — Important Facts:
✅ TRUE: Every figure has 360° as an angle of symmetry (full turn always brings a figure back).
✅ TRUE: If the smallest angle of symmetry is a natural number, it is a factor of 360.
❌ FALSE: Every figure has rotational symmetry (a figure needs an angle LESS than 360° for rotational symmetry).
✅ TRUE: Every figure has 360° as an angle of symmetry (full turn always brings a figure back).
✅ TRUE: If the smallest angle of symmetry is a natural number, it is a factor of 360.
❌ FALSE: Every figure has rotational symmetry (a figure needs an angle LESS than 360° for rotational symmetry).
Exam Practice Questions
Q1. What is a line of symmetry? Give two examples.
Ans: A line that divides a figure into two parts that exactly overlap when the figure is folded along that line is called a line of symmetry (or axis of symmetry). Examples: (1) A square has 4 lines of symmetry. (2) An equilateral triangle has 3 lines of symmetry.
Q2. How many lines of symmetry does a square have? Name them.
Ans: A square has 4 lines of symmetry — (1) Vertical line through midpoints of top and bottom sides, (2) Horizontal line through midpoints of left and right sides, (3) Diagonal from top-left to bottom-right, (4) Diagonal from top-right to bottom-left.
Q3. Does a rectangle (that is not a square) have its diagonal as a line of symmetry? Why?
Ans: No. When a rectangle is folded along its diagonal, the two triangular halves do NOT overlap each other completely. Therefore, the diagonal of a rectangle (non-square) is NOT a line of symmetry.
Q4. What is rotational symmetry? Give an example.
Ans: A figure has rotational symmetry when it looks exactly the same after being rotated by an angle strictly between 0° and 360° about a fixed point (centre of rotation). Example: A square has rotational symmetry with angles 90°, 180°, and 270°.
Q5. Find all the angles of symmetry for an equilateral triangle.
Ans: An equilateral triangle has 3 angles of symmetry: 120°, 240°, and 360°. (Smallest angle = 360° ÷ 3 = 120°. Other angles are multiples: 120°×2 = 240°, 120°×3 = 360°.)
Q6. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry?
Ans: Since 60° is the smallest angle, all other angles are multiples of 60°. So the angles are: 60°, 120°, 180°, 240°, 300°, 360°. The figure has 6 angles of symmetry (order 6).
Q7. Can we have a figure with rotational symmetry whose smallest angle of symmetry is 45°?
Ans: Yes! Since 45° is a factor of 360° (360 ÷ 45 = 8), a figure can have 45° as its smallest angle of symmetry. It would have 8 angles of symmetry: 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°. Example: A regular octagon (8 sides).
Q8. How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?
Ans: The Ashoka Chakra has 24 spokes equally spaced. It has 24 lines of symmetry (one through each spoke and the opposite point). It also has 24 angles of symmetry, with the smallest being 360° ÷ 24 = 15°. The angles are 15°, 30°, 45°, 60°, …, 360°.
Q9. Can a figure have rotational symmetry but no line of symmetry? Give an example.
Ans: Yes! A figure can have rotational symmetry but no line of symmetry. Example: A parallelogram (that is not a rectangle or rhombus) has 180° rotational symmetry but has no line of symmetry. Also, a swastika-like figure (pinwheel/windmill with asymmetric blades) can have rotational but not reflection symmetry.
Q10. How many angles of symmetry does a circle have?
Ans: A circle has infinitely many angles of symmetry. Every angle from 1° to 360° is an angle of symmetry for a circle, because rotating a circle by any angle about its centre always gives back the same circle.
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