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Symmetry – Class 6 Maths Chapter 9 Notes & NCERT PDF

Class 6MathematicsChapter 9NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 9 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 9: Symmetry

Introduction

Look at a flower, a butterfly, a rangoli, a pinwheel, the Taj Mahal or a gopuram. Each has parts that repeat in a definite pattern. A cloud has no such pattern. Chapter 9 turns this beauty into mathematics.

A symmetry refers to a part or parts of a figure that are repeated in some definite pattern.

The chapter's answer key counts 6 lines of symmetry in the flower, 4 in the rangoli and 1 in the butterfly. The pinwheel and the cloud have none.

Key Ideas

9.1 Line of Symmetry

A line that cuts a figure into two parts that exactly overlap when folded along that line is called a line of symmetry (or axis of symmetry).
  • The two parts are mirror halves. A puzzle-piece shape cut down the middle is not symmetric, because the halves don't fit.
  • Many lines: fold a square paper vertically, horizontally and along both diagonals. A square has 4 lines of symmetry.
  • A rectangle that is not a square: its diagonal is not a line of symmetry. Fold it and see. It has only 2 (the vertical and horizontal middle lines).
  • An equilateral triangle (equal sides, equal angles) has 3 lines. A regular hexagon has 6.
  • A triangle can have exactly 0, 1 or 3 lines of symmetry, but never exactly 2.

    • Reflection

    Folding along the line is the same as reflecting each side onto the other. In square ABCD, reflecting in the vertical line swaps A↔B and D↔C. Reflecting in diagonal AC keeps A and C fixed and swaps B↔D. A figure with a line of symmetry has reflection symmetry.

    Making symmetric shapes

  • Ink blot devils: ink on one half of folded paper, press, open. The fold is a line of symmetry.
  • Paper folding and cutting: one cut on folded paper gives a symmetric hole. Two folds (vertical, then horizontal) give a design with 2 lines.
  • Punching game: predict where the holes appear when the paper opens. Every hole has a "partner" reflected across each fold.
  • Kolam designs, and completing half-figures on squared paper so that a blue line becomes a line of symmetry.

    • 9.2 Rotational Symmetry

    A paper windmill has no line of symmetry, yet turning it by 90° about its centre makes it look exactly the same.

    An angle through which a figure can be rotated to look exactly the same is an angle of symmetry. The fixed point is the centre of rotation.
  • Windmill: 90° (quarter turn), 180° (half turn), 270° (three-quarter turn), 360° (full turn), so 4 angles of symmetry.
  • 360° is always an angle of symmetry for every figure.
  • A figure has rotational symmetry only if it has an angle of symmetry strictly between 0° and 360°. The book's strip example matches itself only after a full 360° turn, so it has no rotational symmetry.
  • The order of rotational symmetry is the number of angles of symmetry.

    • Figures with radial arms

  • 4 arms 90° apart: 4 angles of symmetry.
  • 3 arms placed unevenly: only 360°, so no rotational symmetry.
  • 3 arms with equal angles: 360° ÷ 3 = 120° apart, giving angles 120°, 240°, 360°.
  • 5 arms: 72°, 144°, 216°, 288°, 360°. 6 arms: 60°, 120°, … 360°.
  • 7 arms: the smallest angle is 360° ÷ 7 = 51 3/7°, which is not a whole number.
    • The angles of symmetry are always multiples of the smallest one. If the smallest angle is a whole number of degrees, it is a factor of 360.

    Symmetries of a circle

    Rotate a circle by any angle and it matches itself: every angle is an angle of symmetry. Every diameter is a line of symmetry. Fans, wheels and flowers show rotational symmetry too.

    Worked Examples

  • Smallest angle 60°. The others are 120°, 180°, 240°, 300°, 360°, so there are 6 angles in all.
  • 60° is an angle, and exactly two smaller angles exist. The angles below 60° must be 1× and 2× the smallest, and 60° = 3× smallest, so the smallest is 20°.
  • Can the smallest angle be 45°? Yes: 360 ÷ 45 = 8. 17°? No: 17 does not divide 360.
  • Ashoka Chakra: 24 equally spaced spokes, so 24 lines and 24 angles of symmetry (smallest 15°).
  • New Parliament Building (outer boundary): 3 lines of symmetry, with angles 120°, 240°, 360°.
  • Regular polygons (triangle, square, pentagon …) have 3, 4, 5, 6 … lines of symmetry, the counting numbers from 3.

    • Figure it Out: Hints

  • For punching and cutting puzzles, unfold in your head one fold at a time.
  • For "complete the figure", count squares from the blue line and copy each point to the same distance on the other side. Rotate the book for slanting lines.
  • Try This: a quadrilateral with rotational symmetry (180°) but no line of symmetry exists (a slanted, pushed-over rectangle shape).
  • Game (6 × 6 grid): think about how the grid's symmetry can help the second player copy moves.

    • Maths Words

  • Symmetry (सममिति), Symmetrical (सममित)
  • Line of symmetry (सममिति रेखा), Mirror half (दर्पण-अर्ध)
  • Reflection (परावर्तन)
  • Rotation (घूर्णन), Rotational symmetry (घूर्णन सममिति)
  • Centre of rotation (घूर्णन केंद्र)
  • Angle of symmetry (सममिति कोण)
  • Order of rotational symmetry (घूर्णन सममिति का क्रम)
  • Diameter (व्यास)
  • Regular polygon (सम बहुभुज)

    • Common Mistakes

  • Saying a rectangle's diagonal is a line of symmetry. It isn't, unless the rectangle is a square.
  • Forgetting that 360° always counts as an angle of symmetry.
  • Calling a figure "rotationally symmetric" when its only angle is 360°.
  • Mixing up the two kinds: a windmill has rotational symmetry with no lines of symmetry, and a butterfly has one line of symmetry with no rotational symmetry.
  • Claiming a triangle can have exactly 2 lines of symmetry.

    • Maths in Daily Life

    Symmetry is everywhere in Indian design: kolam and rangoli, temple gopurams, the Taj Mahal, the Ashoka Chakra on our flag, and the new Parliament Building. Designers use it for balance, engineers use it in wheels and fans, and nature uses it in flowers and leaves.

    💡 Key Learning Points

    • ✓A figure is symmetrical when its parts repeat in a definite pattern; a cloud is not symmetrical.
    • ✓A line of symmetry folds a figure into two exactly overlapping mirror halves; figures can have one, many or no lines of symmetry.
    • ✓Reflection symmetry means each side of the line is reflected onto the other; shapes can be made by ink blots, folding, cutting and punching.
    • ✓An angle of symmetry is a rotation that makes a figure look the same; 360° is always one, and the others are multiples of the smallest.
    • ✓A figure has rotational symmetry only if it has an angle of symmetry between 0° and 360°; a circle has every angle and every diameter as a symmetry.

    👨‍🏫 Teaching Tips

    • →Start with the chapter's photos (flower, butterfly, rangoli, pinwheel, cloud) and let children point out what repeats.
    • →Give each child a square and a non-square rectangle of paper to fold — let them discover 4 lines vs 2, and that a rectangle's diagonal fails.
    • →Do ink-blot devils and fold-and-cut decorations; ask children to predict the design before opening.
    • →Play the punching game in pairs: one punches, the other predicts where holes appear on unfolding.
    • →Make a cut-out windmill and rotate it over a traced copy with a pin at the centre to find angles of symmetry.
    • →Build radial-arm figures with 3, 4, 5, 6 arms; ask 'what angle must separate the arms?' and connect to 360 ÷ n.
    • →Bring real examples — Ashoka Chakra, kolam, wheel, fan — and classify each by lines and angles of symmetry.
    • →Play the 6×6 domino game and discuss winning strategies using symmetry.

    📋 Assessment Questions

    1. Can the student draw all lines of symmetry of a square, rectangle, equilateral triangle and regular hexagon?
    2. Can the student explain why a triangle cannot have exactly two lines of symmetry?
    3. Can the student list the angles of symmetry of a windmill and decide whether a figure has rotational symmetry?
    4. Can the student find the smallest angle of symmetry for a radial-arm figure with n equal arms (360 ÷ n)?
    5. Can the student decide if 45° or 17° can be a smallest angle of symmetry, giving a reason?

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    Keep learning

    All chapters in Ganita Prakash - Mathematics Class 6

    1. 1. Patterns in Mathematics
    2. 2. Lines and Angles
    3. 3. Number Play
    4. 4. Data Handling and Presentation
    5. 5. Prime Time
    6. 6. Perimeter and Area
    7. 7. Fractions
    8. 8. Playing with Constructions
    9. 9. Symmetry
    10. 10. The Other Side of Zero