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Symmetry

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of symmetry in everyday objects and geometric figures. It explains how a figure is symmetrical when its parts repeat in a definite pattern, either by folding (line/reflection symmetry) or by rotating (rotational symmetry). The one big idea a student must walk away with is that symmetry is about balance and repetition — a figure can have lines of symmetry, angles of rotational symmetry, or both, and understanding these helps us appreciate patterns in nature and art.
Key Concepts (10)
Symmetry
A property of a figure where a part or parts of the figure are repeated in some definite pattern.
e.g. A butterfly has symmetry because its left wing is a mirror image of its right wing.
Line of Symmetry
A line that cuts a plane figure into two parts that exactly overlap when folded along that line.
e.g. A square has 4 lines of symmetry: vertical, horizontal, and two diagonals.
Mirror Halves
The two parts of a figure that exactly cover each other when the figure is folded along a line of symmetry.
e.g. When you fold a triangle along its line of symmetry, one half covers the other half completely.
Reflection Symmetry
A figure that has a line or lines of symmetry is said to have reflection symmetry.
e.g. The Taj Mahal has reflection symmetry because its left and right sides are mirror images.
Rotational Symmetry
A figure has rotational symmetry if it looks exactly the same when rotated by an angle (between 0 and 360 degrees) about a fixed point called the centre of rotation.
e.g. A pinwheel looks the same after a 90° rotation about its centre.
Centre of Rotation
The fixed point about which an object is rotated to check for rotational symmetry.
e.g. For a windmill, the red dot at the centre is the centre of rotation.
Angle of Symmetry (Rotational)
An angle through which a figure can be rotated to look exactly the same as the original.
e.g. For a square, the angles of symmetry are 90°, 180°, 270°, and 360°.
Radial Arms
Lines or arms radiating from a central point in a figure, used to study rotational symmetry.
e.g. A figure with 4 radial arms has angles of symmetry that are multiples of 90°.
Order of Rotational Symmetry
The number of distinct positions (angles) in which a figure looks exactly the same during a full 360° rotation.
e.g. A square has order 4 because it matches itself at 90°, 180°, 270°, and 360°.
Asymmetrical
A figure that does not have any symmetry — no line of symmetry and no rotational symmetry (except 360°).
e.g. A picture of clouds is asymmetrical because there is no repetitive pattern.
Formulas & Equations (2)
Smallest angle of symmetry = 360° ÷ number of radial arms (or order of symmetry)
Reads as: Smallest angle of symmetry equals 360 degrees divided by the number of radial arms or the order of symmetry.
Variables: Smallest angle of symmetry (in degrees), number of radial arms (dimensionless), order of symmetry (dimensionless)
Use when: When finding the smallest angle of symmetry for a figure with radial arms or a regular polygon.
⚠ Students often forget that the smallest angle must be a factor of 360; if 360 is not divisible, the figure does not have rotational symmetry (except 360°).
Angles of symmetry = multiples of the smallest angle of symmetry (up to 360°)
Reads as: Angles of symmetry are the multiples of the smallest angle of symmetry, up to 360 degrees.
Variables: Angles of symmetry (in degrees), smallest angle of symmetry (in degrees)
Use when: When listing all angles of symmetry for a figure once the smallest angle is known.
⚠ Students often forget to include 360° as an angle of symmetry — every figure has 360° as an angle of symmetry.
How It Works
Finding Lines of Symmetry by Folding
  1. Take a paper figure (e.g., a square).
  2. Fold it along a possible line (vertical, horizontal, or diagonal).
  3. Check if the two halves overlap exactly.
  4. If they do, that fold line is a line of symmetry.
  5. Repeat for all possible folds to find all lines of symmetry.
💡 A diagonal of a rectangle (non-square) is NOT a line of symmetry — the two halves do not overlap.
Generating Symmetric Shapes by Ink Blot Method
  1. Take a piece of paper and fold it in half.
  2. Open the paper and spill a few drops of ink or paint on one half.
  3. Press the halves together and then open the paper again.
  4. The resulting figure is symmetric, with the fold line as the line of symmetry.
💡 The resulting figure has only one line of symmetry — the fold line.
Generating Symmetric Shapes by Paper Folding and Cutting
  1. Fold a sheet of paper (once or multiple times).
  2. Make a cut along a dotted line or shape on the folded paper.
  3. Unfold the paper to reveal a symmetric pattern.
  4. The fold lines become lines of symmetry in the final figure.
💡 If you fold the paper multiple times, the final figure will have multiple lines of symmetry corresponding to each fold.
Determining Rotational Symmetry by Rotation
  1. Mark the centre of rotation on the figure.
  2. Rotate the figure by a small angle (e.g., 90°).
  3. Check if the rotated figure looks exactly the same as the original.
  4. If yes, that angle is an angle of symmetry.
  5. Continue rotating by the same angle until you complete 360° to find all angles of symmetry.
💡 360° is always an angle of symmetry for every figure — it is the full turn back to the original position.
Finding Angles of Symmetry for Radial Arms Figures
  1. Count the number of radial arms (e.g., 3, 4, 5).
  2. Divide 360° by the number of arms to get the smallest angle of symmetry.
  3. List all multiples of this smallest angle up to 360°.
  4. These multiples are the angles of symmetry.
💡 For a figure with 3 radial arms to have rotational symmetry, the angles between adjacent arms must be equal (120° each). If they are not equal, the figure does not have rotational symmetry (except 360°).
Exam Traps — Don't Make These Mistakes
⚠ Diagonal of a Rectangle is a Line of Symmetry
✗ The diagonal of a rectangle is a line of symmetry because it cuts the rectangle into two equal triangles.
→ Although the two triangles are equal in area, they are not mirror images — when folded along the diagonal, the two halves do not overlap exactly. The diagonal of a rectangle (non-square) is NOT a line of symmetry.
✓ The diagonal of a rectangle is NOT a line of symmetry. Only the vertical and horizontal lines through the centre are lines of symmetry for a rectangle.
⚠ Every Figure with Radial Arms has Rotational Symmetry
✗ A figure with 3 radial arms always has rotational symmetry with angles 120°, 240°, and 360°.
→ A figure with 3 radial arms only has rotational symmetry if the angles between adjacent arms are equal (120° each). If the arms are unevenly spaced, the figure does not have rotational symmetry (except 360°).
✓ A figure with 3 radial arms has rotational symmetry only if the angle between adjacent arms is 120°. Otherwise, it has no rotational symmetry.
⚠ All Symmetric Figures have Both Line and Rotational Symmetry
✗ If a figure has a line of symmetry, it must also have rotational symmetry.
→ Some figures have line symmetry but no rotational symmetry (e.g., an isosceles triangle has one line of symmetry but only 360° as an angle of symmetry). Some figures have rotational symmetry but no line symmetry (e.g., a pinwheel).
✓ A figure may have line symmetry without rotational symmetry, or rotational symmetry without line symmetry, or both.
⚠ 360° is Not Always an Angle of Symmetry
✗ Some figures do not have 360° as an angle of symmetry because they do not look the same after a full turn.
→ Every figure, when rotated by 360°, returns to its original position. Therefore, 360° is always an angle of symmetry for every figure.
✓ 360° is always an angle of symmetry for any figure because a full rotation brings the figure back to its original position.
⚠ Smallest Angle of Symmetry Must be a Whole Number
✗ If 360° divided by the number of arms gives a fraction, the figure cannot have rotational symmetry.
→ The smallest angle of symmetry can be a fraction (e.g., 51 3/7° for 7 arms). The figure still has rotational symmetry if the arms are equally spaced. The condition is that the angle must be a factor of 360 only if we require whole number degrees.
✓ The smallest angle of symmetry is 360° divided by the number of equally spaced radial arms. It can be a fractional degree, and the figure still has rotational symmetry.
Compare & Contrast
Type of transformation
Line of Symmetry (Reflection Symmetry)
Rotational Symmetry
Line symmetry involves folding the figure along a line so that the two halves overlap exactly. Rotational symmetry involves rotating the figure about a point so that it looks the same at certain angles.
Number of lines of symmetry
Square
Rectangle (non-square)
A square has 4 lines of symmetry (vertical, horizontal, and two diagonals). A rectangle (non-square) has only 2 lines of symmetry (vertical and horizontal) — its diagonals are NOT lines of symmetry.
Exam Focus
Marks pattern: 2-3 MCQs (1 mark each), 1-2 Very Short Answer questions (1-2 marks), 1 Short Answer question (3 marks), 1 Long Answer question (5 marks) — total approximately 10-12 marks
Most asked: Identifying lines of symmetry in given figures · Finding angles of symmetry and order of rotational symmetry · Distinguishing between reflection and rotational symmetry · Completing figures to have a given line of symmetry · Determining if a figure has rotational symmetry and listing its angles
Don't skip: Questions that ask to find all lines of symmetry and all angles of symmetry for a given figure (e.g., square, regular polygon, or a composite shape); Questions that require completing a figure on squared paper to have one or two lines of symmetry; Questions that ask to draw a triangle/quadrilateral with specific symmetry properties
One-Line Revision (read aloud before the exam)
  1. Symmetry means a part of a figure repeats in a definite pattern.
  2. A line of symmetry divides a figure into two mirror halves that exactly overlap when folded.
  3. A square has 4 lines of symmetry: vertical, horizontal, and two diagonals.
  4. A rectangle (non-square) has only 2 lines of symmetry: vertical and horizontal.
  5. A diagonal of a rectangle is NOT a line of symmetry.
  6. Rotational symmetry means a figure looks the same after being rotated by an angle less than 360°.
  7. The centre of rotation is the fixed point about which the figure is rotated.
  8. The smallest angle of symmetry for a figure with n equally spaced radial arms is 360°/n.
  9. Angles of symmetry are always multiples of the smallest angle of symmetry.
  10. 360° is always an angle of symmetry for every figure.
  11. A figure can have line symmetry but no rotational symmetry, or rotational symmetry but no line symmetry.
  12. A circle has infinite lines of symmetry and every angle is an angle of symmetry.
  13. An isosceles triangle has exactly one line of symmetry and no rotational symmetry (except 360°).
  14. An equilateral triangle has 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
  15. The order of rotational symmetry is the number of distinct angles (including 360°) at which the figure looks the same.

MCQs — 10 Questions with Answers

Q1. A line that divides a figure into two parts that exactly overlap when folded is called a _____ of symmetry.

  • A. line ✓
  • B. fold
  • C. mirror
  • D. edge

Answer: A — A line of symmetry is the technical term for the fold line where two identical mirror halves meet.

Q2. How many lines of symmetry does a square have?

  • A. 1
  • B. 2
  • C. 4 ✓
  • D. Infinite

Answer: C — A square has 4 lines of symmetry: one vertical, one horizontal, and two through opposite corners (diagonals).

Q3. Which of these shapes has only 2 lines of symmetry?

  • A. Square
  • B. Rectangle ✓
  • C. Circle
  • D. Equilateral triangle

Answer: B — A non-square rectangle has exactly 2 lines of symmetry: vertical and horizontal, but the diagonals are NOT lines of symmetry.

Q4. In an ink blot activity, you fold paper, add ink, and unfold. What do you get?

  • A. Two different ink patterns
  • B. A symmetric pattern with a line of symmetry at the fold ✓
  • C. A random asymmetric shape
  • D. Only one ink blot on the paper

Answer: B — The fold line becomes the line of symmetry, so the ink pattern on one half is mirrored perfectly on the other half.

Q5. Is the diagonal of a rectangle (that is not a square) a line of symmetry?

  • A. Yes, always
  • B. No, the two halves do not overlap exactly ✓
  • C. Only if the rectangle is very long
  • D. Yes, if we fold it carefully

Answer: B — When you fold a non-square rectangle along its diagonal, the two parts do not match perfectly, so it is not a line of symmetry.

Q6. Which Indian structure shows clear line symmetry when viewed from the front?

  • A. A cloud
  • B. A tree
  • C. Taj Mahal ✓
  • D. A river

Answer: C — The Taj Mahal has vertical line symmetry with its dome and building design mirrored on both sides.

Q7. When you fold a square piece of paper along its diagonal and unfold it, the crease is a line of symmetry. How many such diagonal lines of symmetry can a square have?

  • A. 1
  • B. 2 ✓
  • C. 4
  • D. 0

Answer: B — A square has 2 diagonal lines of symmetry: one from corner A to corner C, and another from corner B to corner D.

Q8. Which of the following figures has NO line of symmetry?

  • A. A butterfly
  • B. A flower
  • C. A cloud ✓
  • D. A square

Answer: C — Clouds are asymmetrical—there is no line along which you can fold a cloud to get two identical mirror halves.

Q9. In the paper folding and cutting activity, you fold paper twice (once vertically, once horizontally) and make one cut. When you unfold, how many copies of your cut shape appear due to symmetry?

  • A. 2 copies
  • B. 3 copies
  • C. 4 copies ✓
  • D. 1 copy

Answer: C — Two folds create two lines of symmetry (vertical and horizontal), so the cut shape is mirrored in all 4 quadrants when unfolded.

Q10. A butterfly has a body with one wing on the left and one on the right. The left wing is a mirror image of the right wing. This means the butterfly has _____ line(s) of symmetry.

  • A. 0
  • B. 1 ✓
  • C. 2
  • D. 4

Answer: B — A butterfly has exactly 1 vertical line of symmetry running down its body, with the left and right wings being mirror images.

Flashcards

What is a line of symmetry?

A line that cuts a figure into two parts that exactly overlap when the figure is folded along that line.

How many lines of symmetry does a square have?

A square has 4 lines of symmetry: one vertical, one horizontal, and two diagonal lines.

How many lines of symmetry does a rectangle (not a square) have?

A rectangle has 2 lines of symmetry: one vertical and one horizontal, but NOT along the diagonals.

Is the diagonal of a non-square rectangle a line of symmetry?

No, when you fold a rectangle along its diagonal, the two halves do not overlap exactly.

What is reflection symmetry?

When one side of a figure is the mirror image of the other side, reflected across a line of symmetry.

Name two Indian structures that have line symmetry.

Taj Mahal (dome and building design) and gopurams (temple towers) both show left-right mirror symmetry.

What happens when you fold paper along a line of symmetry and cut it?

When you unfold the paper, the cut-out pattern appears symmetrically on both sides of the fold line.

Does a cloud have line symmetry?

No, clouds do not have line symmetry because there is no line that divides them into two identical mirror halves.

How many lines of symmetry does a butterfly have?

A butterfly has 1 line of symmetry running vertically down the middle of its body.

What is the difference between a symmetrical figure and an asymmetrical figure?

A symmetrical figure has at least one line of symmetry where two parts are mirror halves, while an asymmetrical figure has no such line.

Important Board Questions

What is a line of symmetry? Give one example from your daily life. [1 mark]

Define as a fold line dividing a figure into two identical mirror halves. Example: butterfly wings, your face, or a rangoli design.

How many lines of symmetry does a square have? Name them or draw and label them. [2 marks]

List 4 lines: vertical, horizontal, and two diagonals. A diagram showing all 4 fold lines is helpful.

Why is the diagonal of a rectangle (not a square) NOT a line of symmetry? Explain with a folding test. [3 marks]

Fold a rectangle along its diagonal. The two halves do not overlap exactly because the rectangle has unequal sides. Compare this to a square where the diagonal IS a line of symmetry.

Describe the ink blot activity step by step. Explain why the resulting pattern has a line of symmetry and identify where this line is located. [5 marks]

Steps: fold paper, add ink to one half, press and open. The fold line is the line of symmetry. Every point on the left is mirrored on the right. Include a simple sketch showing the fold line and symmetric pattern.

True or False: A cloud is a symmetric figure. Give a reason for your answer. [2 marks]

False — there is no line along which you can fold a cloud to get two identical mirror halves. Clouds are asymmetrical and have no line of symmetry.

Assertion-Reason Questions for Symmetry

Q1. Assertion (A):

A square has 4 lines of symmetry.

Reason (R):

A square can be folded in 4 different ways so that both halves overlap completely: vertically, horizontally, and along both diagonals.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why a square has exactly 4 lines of symmetry. The four folds (vertical, horizontal, and two diagonals) each create a perfect mirror image.

Q2. Assertion (A):

A cloud is an example of a symmetrical figure.

Reason (R):

Clouds have a random, uneven shape with no repeating pattern.

Answer: D — A is false but R is true — Both the assertion and reason are false. Clouds are asymmetrical (not symmetrical) because they do not have a repeating pattern or a line of symmetry. The reason correctly describes clouds but does not support the assertion.

Q3. Assertion (A):

When you fold paper along a line of symmetry and punch a hole, the holes appear in symmetrical positions when unfolded.

Reason (R):

The fold line acts as a line of symmetry, so anything marked on one side appears mirrored on the other side.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the assertion. The holes are symmetrically placed because the fold creates reflection symmetry — whatever is on one side gets reflected to the other side.

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Case-Based Questions (CBQ) for Symmetry

Case Study:

Priya is helping her mother decorate the house for Diwali. They want to make a paper decoration by folding a rectangular piece of red paper in half vertically, then cutting out a shape along the fold. When they unfold the paper, they see a beautiful symmetrical design. Priya notices that the left side of the design looks exactly like the right side.

Q1. What is the line of symmetry in Priya's paper decoration?

  • A. The vertical fold line in the middle ✓
  • B. The horizontal line through the paper
  • C. The diagonal from corner to corner
  • D. There is no line of symmetry

Answer: A

Q2. Why does the left side of the design look exactly like the right side?

  • A. Because Priya cut carefully on both sides
  • B. Because the fold line is a line of symmetry, creating mirror halves ✓
  • C. Because the paper is red
  • D. Because she unfolded the paper slowly

Answer: B

Q3. If Priya folds the paper again along the same vertical line, what will happen?

  • A. The left half will cover the right half completely ✓
  • B. The paper will tear
  • C. The design will change
  • D. Nothing will change because the paper is already symmetrical

Answer: A

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