Symmetry | CBSE Class 6 Maths Notes

This note covers repeated patterns, matching halves, folding and reflection, completing drawings, turning figures, angles and centres of rotation, radial arms, circles, and symmetric designs made with paper and tiles.

What is symmetry, and how can a fold reveal it?

Symmetry occurs when parts of a figure repeat in a definite pattern. A figure with such a pattern is called symmetrical. Looking for the repeated part is a useful first step, but explaining how the parts match makes the observation precise.

In pictures of a flower, butterfly, rangoli and pinwheel, it appears that some parts repeat, and these repetitions seem to follow a definite pattern. In the photograph of clouds, there is no such repetitive pattern. This comparison concerns those particular pictures.

What are mirror halves?

Mirror halves are two parts that cover each other completely when the figure is folded along the line separating them. A line of symmetry, also called an axis of symmetry, is a line along which this exact matching occurs.

Definition: A line of symmetry divides a flat figure into two parts that overlap exactly when folded along that line.

The important test is complete overlap. A line through the middle of a drawing does not automatically pass this test. Look at the outline and at the details within the drawing, then decide whether one side would cover the other.

A blue triangle has a vertical dotted line, a pink circle at its top and yellow circles at its lower corners. Beside it, four orange puzzle pieces have a dotted line through their middle.

Reference: NCERT Class 6, p. 219, figures a and b

Folding the triangle picture along its dotted line gives matching halves. Folding the puzzle-piece picture along its dotted line does not make its two sides fit exactly. The dotted line is therefore a symmetry line for the triangle picture, but not for the puzzle-piece picture.

When checking a proposed line, trace or fold the figure if visual inspection is uncertain. The reason for accepting the line should be the matching of the whole figure across it, rather than its position on the page.

How do square folds explain reflection symmetry?

A square has four equal sides and four right angles, which are square-corner angles. It has more than one line of symmetry. A line segment is a straight part of a line between two endpoints. A diagonal is a segment joining opposite corners. The square's diagonals, as well as its central vertical and horizontal lines, give matching folds.

Property: A square has four lines of symmetry

1. Fold a square sheet vertically into equal halves.
2. Fold it horizontally into equal halves, then open the folds.
3. Fold along one diagonal, check the overlap, and open it.
4. Fold along the other diagonal and open the sheet again.

Vertical means up and down on the page; horizontal means across the page. These descriptions help name the folds, but symmetry depends on matching parts. Turning the sheet changes its position without changing which folds make its halves overlap.

Worked example 1. Find the lines of symmetry of a square using its folds.

Answer: The central vertical fold, central horizontal fold and 2 diagonal folds all give complete overlap. There are 4 lines of symmetry altogether.

How do labelled corners move under reflection?

Reflection describes one side being mirrored across a line to the other side. A figure with at least one line of symmetry has reflection symmetry. Label a square's corners A at the top left, B at the top right, C at the bottom right and D at the bottom left.

The square has corners A, B, C and D in the positions described above. A vertical line through the middle separates its left and right halves.

Reference: NCERT Class 6, p. 222

A rectangle has four right angles. For a rectangle that is not a square, folding along a diagonal does not make the halves overlap exactly. Therefore, the result about a square's diagonal cannot be transferred to every rectangle.

How can ink, folds and holes create symmetrical patterns?

Folding can generate a symmetric design as well as test one. In an ink-blot activity, pressing folded halves together transfers ink or paint from one half to the other. The fold provides the line across which the resulting parts match.

How is an ink-blot pattern made?

1. Take a sheet of paper and fold it in half.
2. Open it and place a few drops of ink or paint on one half.
3. Bring the halves together and press them against each other.
4. Open the paper and examine the pattern on both sides of the fold.

Check the fold as a symmetry line. Then investigate whether another line also gives matching halves. Making the pattern with one fold does not by itself establish the total number of lines of symmetry in the finished design.

What happens when folded paper is cut or punched?

In paper cutting, a cut through folded layers produces related parts when the sheet opens. Predict the unfolded shape first, then make the cut and compare the result with the prediction. Repeated folds can produce decorative patterns with repeated cut-outs.

The sequence shows a folded sheet, a hole through the folded paper, and an opened square with two holes at the same height on opposite sides of its middle.

Reference: NCERT Class 6, p. 223

Worked example 2. A square sheet is folded vertically in half. A hole is punched through both layers away from the fold. Explain the paired holes visible after unfolding.

Answer: There are 2 holes, one on each side of the fold. Folding again makes them overlap, so the vertical crease is a line of symmetry of this punched pattern.

For a pattern with several holes, test the entire arrangement. A proposed fold must bring every hole to its matching partner while also matching the sheet's boundary. Finding one matching pair is not enough if another part fails to match.

When several folds were used, open them one at a time in your reasoning. Each unfolding reveals how the cut or hole is repeated across that particular crease. This helps connect the folded action with the completed design.

How do you complete a drawing with a given symmetry line?

A grid is a network of regularly spaced lines; squared paper has a grid of equal squares. A partial drawing and a marked symmetry line provide enough guidance to construct its reflected part. The missing part must fit the given part when folded.

What should be matched across the line?

1. Identify the given line of symmetry before adding any part of the drawing.
2. Locate the corners and changes of direction in the existing outline.
3. Place their matching points across the line, checking the fold relationship.
4. Join the new points in the corresponding order, then test the completed outline.

For a vertical line, the matching part lies on the other side to the left or right. For a horizontal line, it lies above or below. A slanting line requires the same folding test; rotating the paper can help you see the relationship.

Corresponding points are points that match under the fold. On opposite sides of a symmetry line, they are equally far from it, measured straight across the line. Points on the symmetry line remain in the same position during the fold.

How do two given lines change the task?

When two lines are prescribed, the final drawing must satisfy both. Reflect the supplied part, then check the additional parts needed across the other line. Do not stop when the picture matches across just one of the two lines.

The same care applies to coloured tiles. A tile is a small piece used to build a larger pattern. The direction of its coloured parts matters: matching square outlines alone does not show that the coloured design has reflection symmetry.

Note: Completing a drawing means preserving the given part and adding its matching parts. Changing the supplied outline to make an easier pattern answers a different task.

For a requested number of symmetry lines, count only the lines that pass the whole-pattern test. A design asked to have exactly two lines must work across both of them, and checking possible additional folds is part of verifying the result.

What is rotational symmetry, and why is a full turn insufficient?

Rotation means turning a figure about a fixed point. That fixed point is its centre of rotation. The amount of turning is an angle, measured here in degrees. The symbol ° means degrees; a full turn is 360°.

An angle of symmetry, or angle of rotational symmetry, is a turn that brings the figure into exact overlap with its starting position. Keep the centre fixed during the test. Moving the figure across the page is not the same operation as turning it about that point.

Property: A matching turn before 360° establishes rotational symmetry

Definition: A figure has rotational symmetry when at least one angle of symmetry is strictly between 0° and 360°.

A turn of 0° means no turn. Every figure returns to its starting position after 360°, so that angle alone cannot distinguish figures with rotational symmetry. There must also be a match after a positive turn smaller than a full turn.

The illustrated paper windmill has no line of symmetry, but it matches itself after a quarter turn about its centre. A quarter turn is 90°, a half turn is 180°, and a three-quarter turn is 270°.

Four green blades meet at a red central point. Turning the windmill through 90° about this point restores the same appearance, although folding it does not produce matching halves.

Reference: NCERT Class 6, p. 230

Worked example 3. A windmill design first matches after 90° and matches again after each further quarter turn. List its angles of symmetry in one full turn.

Answer: Its angles are 90°, 180°, 270° and 360°. The first three lie below a full turn, so the design has rotational symmetry.

For a figure with finitely many matching positions, its order of rotational symmetry is the number of these positions during one full turn, counting 360° and excluding 0°. The windmill therefore has order 4.

How do a square and a strip behave when rotated?

A square shows that one figure can have both reflection and rotational symmetry. Its folding lines were tested earlier. To investigate its turns, keep its centre fixed and compare the square with its original position after each quarter turn.

How do the square's corners change position?

Use the same corner labels: A at the top left, B at the top right, C at the bottom right and D at the bottom left. A clockwise turn goes in the direction of a clock's hands.

After a clockwise 90° turn, A reaches B's former position, B reaches C's, C reaches D's and D reaches A's. The corner labels help track the motion. The square's outline still overlaps its original outline completely.

Worked example 4. Find the angles of symmetry of an unmarked square rotated about its centre.

Answer: The square overlaps itself after 90°, 180°, 270° and 360°. These give 4 matching positions in one full turn, so its order of rotational symmetry is 4.

Why does the illustrated strip fail the rotational test?

The strip has a shorter top edge and a longer bottom edge, joined by sloping sides. After a half turn, the longer edge is above and the shorter edge is below. The turned strip therefore does not cover the original strip exactly.

The first strip widens towards its lower edge and has a black point inside. The second drawing, after 180°, widens towards its upper edge.

Reference: NCERT Class 6, p. 232

Worked example 5. The strip described above matches its original position only after a full turn. Does it have rotational symmetry?

Answer: Its only angle of symmetry in one full turn is 360°. Since no matching angle lies strictly between 0° and 360°, it does not have rotational symmetry.

This comparison separates an angle of symmetry from having rotational symmetry. The strip has the full-turn angle, as every figure does, but it fails the additional requirement. A square meets that requirement at several smaller angles.

How do radial arms produce matching turns?

Radial arms are arms extending outwards from a common centre. Two neighbouring arms are called adjacent arms. Both the shapes of the arms and the angles between them affect whether turning the figure produces an exact match.

A cross made from four matching arms with 90° between adjacent central lines matches after each quarter turn. However, simply counting arms is insufficient. The illustrated three-arm arrangement with unequal gaps returns to itself only after a full turn.

Why must the gaps match in the symmetric three-arm design?

Suppose the three matching arms are to exchange places in turn. Label the three central gaps A, B and C. The notation ∠A means the angle labelled A; ∠B and ∠C name the other two angles. The symbol = means “is equal to”.

For the turned figure to overlap the original, ∠A must fit ∠B, ∠B must fit ∠C, and ∠C must fit ∠A. Therefore, ∠A = ∠B = ∠C. The full turn is shared equally among the three gaps.

The symbol ÷ means “divided by”. Thus, 360° ÷ 3 = 120°. Each gap must be 120°. Three equally spaced matching arms then give symmetry angles of 120°, 240° and 360°.

Rough diagrams label the central gaps A, B and C. A sequence below shows the initial position and positions after 120°, 240° and 360°. Colours are added to show the rotations.

Reference: NCERT Class 6, p. 234

Worked example 6. Three matching radial arms are equally spaced around their centre. Find the gap between adjacent arms and list the matching turns.

Answer: Divide the full turn equally: 360° ÷ 3 = 120°. The matching turns are 120°, 240° and 360°, giving 3 angles of symmetry in one full turn.

A useful practical check uses two copies. Keep one drawing fixed, cut out the other, and turn the cut-out over the fixed drawing. At a proposed symmetry angle, every arm must cover the matching arm beneath it, not merely point in a similar direction.

How can you calculate and check angles of symmetry?

The smallest angle of symmetry is the least positive turn that produces a match, when such a least turn exists. A multiple is obtained by multiplying a number by a whole number. Repeating the smallest matching turn gives successive matching positions.

Property: Finite matching positions divide a full turn into equal steps

For the finite rotational patterns considered here, divide 360° by the number of matching positions to find the smallest angle. Then list successive multiples of this angle up to and including 360°. Do not count the unchanged starting position as another angle.

Worked example 7. Make a design from five matching radial arms equally spaced about their centre. Find its angles of symmetry.

Answer: The smallest matching turn is 360° ÷ 5 = 72°. Successive matching turns are 72°, 144°, 216°, 288° and 360°, giving exactly 5 angles.

Worked example 8. A figure has smallest angle of symmetry 60°. Find its other angles of symmetry.

Answer: Continue in steps of 60° up to a full turn. The other angles are 120°, 180°, 240°, 300° and 360°. Including 60°, there are 6 matching angles.

What if the smallest angle is not a whole number?

A whole number is zero or a positive counting number. With exactly seven equally spaced matching positions, the smallest turn is 360° ÷ 7 = 51 3/7°. Here 51 3/7 is a mixed fraction, meaning 51 and three sevenths. A valid angle need not be a whole number of degrees.

Worked example 9. Can a figure have smallest angle of symmetry 45° or 17°?

Answer: 45° works because 360° ÷ 45° = 8, a whole number of equal steps. 17° cannot be the smallest angle because it does not divide 360° into a whole number of steps.

A natural number is a positive counting number; a factor divides a whole number exactly. If the smallest angle is a natural number of degrees, that number must be a factor of 360. This condition does not require every smallest angle to be a natural number.

Why is a circle a special case?

A circle is a boundary whose points are equally far from its centre. Its rim means that boundary. Unlike a square or a finite radial-arm pattern, an unmarked circle matches itself after a turn through any angle about its centre.

Does a circle have a smallest positive matching turn?

For most figures there is a smallest angle of symmetry, except for the most symmetric shapes like the circle. A circle has no smallest positive angle of symmetry: whatever positive angle is proposed, a smaller positive turn also gives a match.

This is why the equal-step calculations for a finite number of matching positions should not be used to assign a finite order to the circle. Its matches are not restricted to the particular angle lists for two, three or four positions.

How do diameters give reflection symmetry?

A diameter is a straight segment through the centre joining two points on the circle. Join a point on the rim to the centre and continue the segment to the opposite rim. The line containing that diameter divides the circle into matching halves.

Definition: Every diameter of a circle gives a line of reflection symmetry, and every angle gives a rotational match about the centre.

Choose another point on the rim and repeat the construction. The same folding relationship holds. The circle therefore has infinitely many lines of symmetry, meaning that their number is not limited to any finite count.

Can colouring change the symmetry of a circular design?

A sector is a wedge-shaped part of a circle between two segments from the centre and the intervening rim. When sectors are coloured, test the colours as part of the figure. The circular boundary can match even when the complete coloured design does not.

To make a sector design with three or four matching positions, arrange the colours to repeat at the corresponding turns. Check the complete coloured pattern after turning it. Properties of the unmarked circle do not automatically describe every decorated circular picture.

How should reflection and rotation be compared in designs?

Reflection symmetry asks whether a fold gives matching halves. Rotational symmetry asks whether a turn smaller than a full turn restores the whole figure. The tests use different actions, so a result from one test does not settle the other.

How can these tests guide practical activities?

A triangle is a three-sided figure. Drawing triangles with one, three or no symmetry lines gives practice in the folding test. An equilateral triangle, with three equal sides and equal angles, has three symmetry lines. Each joins a corner to the middle of the opposite side.

For a design with curved boundaries, use the same exact-overlap rule. A curve does not remove the possibility of symmetry. Likewise, a shape drawn at a slant should be tested by its own folds and turns, rather than by the page's edges.

In tile activities, use the given coloured tiles to complete a pattern with exactly two symmetry lines. Another task uses 16 tiles to make designs with exactly one or exactly two lines. Test the internal colour arrangement along with the outer boundary.

What does the grid game ask you to investigate?

The game uses a 6 by 6 grid, meaning six rows and six columns of squares. Two players take turns drawing a line to cover two adjacent squares. Here adjacent squares are neighbours sharing a side.

Each line may be horizontal or vertical, and lines cannot overlap. The player unable to place another line loses. This is an opportunity to investigate arrangements and strategies; a proposed strategy must respect the allowed moves and avoid squares already covered.

Across these activities, the useful habit is to identify the object being tested. It may be an outline, a coloured arrangement or a complete drawing. State the fold or centre, carry out the appropriate action, and check all the relevant parts.

Glossary

• Symmetry — Repetition of parts of a figure in a definite pattern that can be recognised and tested.
• Line of symmetry — A line along which a figure folds into two parts that overlap completely.
• Mirror halves — Two parts of a figure that match completely when folded along their separating symmetry line.
• Reflection symmetry — The property of a figure having at least one line that gives matching folded halves.
• Diagonal — A straight segment joining opposite corners of a square or rectangle.
• Rotation — The turning of a figure about a fixed point while that point stays in place.
• Centre of rotation — The fixed point about which a figure turns during a test for rotational symmetry.
• Angle of symmetry — An angle through which a figure turns to overlap its original position exactly.
• Rotational symmetry — A figure's property of matching itself after a turn strictly between zero and 360 degrees.
• Order of rotational symmetry — The finite number of matching positions in one full turn, including 360 degrees but excluding zero.
• Radial arms — Parts of a figure that extend outwards from a common central point.
• Smallest angle of symmetry — The least positive turn that restores a figure's appearance, when such a least angle exists.
• Diameter — A straight segment passing through a circle's centre and joining two points on its rim.
• Sector — A wedge-shaped part of a circle between two segments from its centre and the intervening rim.

Common errors and misconceptions

• Misconception: Any line through a figure's middle is a symmetry line. Correct: Folding along it must make the two parts overlap completely.
• Misconception: A square has only vertical and horizontal symmetry lines. Correct: Both diagonals also work, giving four lines altogether.
• Misconception: Every rectangle's diagonal is a symmetry line. Correct: A rectangle that is not a square does not fold into matching halves along its diagonal.
• Misconception: Matching after 360° proves rotational symmetry. Correct: Every figure does that; rotational symmetry requires an additional match strictly between 0° and 360°.
• Misconception: A figure without reflection symmetry cannot have rotational symmetry. Correct: The illustrated windmill has no symmetry line but matches after quarter turns.
• Misconception: Three radial arms automatically give three angles of symmetry. Correct: Matching arms must also have the appropriate equal spacing for that pattern.
• Misconception: Every figure has a smallest positive angle of symmetry. Correct: Most figures do, but a circle matches at every angle and has no smallest positive one.
• Misconception: A circular outline makes every colouring rotationally symmetric at every angle. Correct: The colours must also match when the complete design is turned.

Exam-style questions with model answers

Q1. What is a line of symmetry? State the test for deciding whether a proposed line is one. [2 marks]

1. A line of symmetry divides a figure into two parts that can match completely.
2. Fold along the proposed line: it qualifies when one part overlaps the other exactly.

Q2. A square has corners A at the top left, B at the top right, C at the bottom right and D at the bottom left. State where each corner goes under reflection in the central vertical line. [4 marks]

1. A goes to B's original position: the upper left corner is reflected to the upper right corner.
2. B goes to A's original position: the upper right corner is reflected to the upper left corner.
3. C goes to D's original position: the lower right corner is reflected to the lower left corner.
4. D goes to C's original position: the lower left corner is reflected to the lower right corner.

Q3. A windmill design first matches itself at 90° about its fixed centre and matches after every further quarter turn. List its angles in one full turn, give its order, and explain whether it has rotational symmetry. [3 marks]

1. The matching angles are 90°, 180°, 270° and 360°. Each quarter turn adds another 90° to the total rotation.
2. Its order is 4 because four matching positions occur in one full turn, counting 360° but excluding the unchanged starting position.
3. It has rotational symmetry because matching turns occur strictly between 0° and 360°, including the first match at 90°.

Q4. A strip matches its starting position only after a full turn. State its angle of symmetry in one turn and explain whether it has rotational symmetry. [2 marks]

1. Its angle of symmetry is 360°, when it returns to its starting position.
2. It has no rotational symmetry because there is no matching turn strictly between 0° and 360°.

Q5. Three identical radial arms are equally spaced around a fixed centre. Explain why the gaps are equal, calculate their size, list the angles of symmetry, state the order, and describe a cut-out check. [5 marks]

1. The matching arms exchange positions when the figure turns. For each arm to cover the next arm exactly, the angular gaps between adjacent arms must match.
2. A full turn of 360° is shared equally among three gaps. Dividing gives 360° ÷ 3 = 120° for each gap.
3. The angles of symmetry are 120°, 240° and 360°, obtained by taking successive turns of 120° about the same fixed centre.
4. The order is 3 because these are three matching positions during one complete turn, including the final return at 360°.
5. Trace and cut out a second copy, keep the original drawing fixed, and rotate the cut-out about the centre to check complete overlap at each listed angle.

Q6. A figure has 60° as its smallest angle of symmetry. State the equal-step rule, calculate the number of matching positions, list the other angles, explain the rotational-symmetry decision, state how 360° is counted, and give a check on the word “smallest”. [6 marks]

1. Its matching positions occur in equal angular steps. Starting with the smallest matching angle, repeat turns of 60° until the figure completes one full turn.
2. The number of matching positions is 360° ÷ 60° = 6, since six equal steps fill the full turn.
3. The other angles of symmetry are 120°, 180°, 240°, 300° and 360°. These follow the initial match at 60°.
4. The figure has rotational symmetry because 60° is positive and smaller than 360°, so a match occurs before a complete turn.
5. Count 360° once as the final matching position in the full turn. Do not add 0° as a seventh position.
6. Calling 60° the smallest means no positive turn below 60° restores the complete figure. A smaller successful turn would contradict the given information.

Q7. Compare an unmarked square and an unmarked circle, each rotated about its centre: give the square's matching angles, the circle's matching angles, and whether the circle has a smallest positive matching angle. [3 marks]

1. The square matches after 90°, 180°, 270° and 360°. These are its four angles of symmetry within one full turn.
2. The circle matches after every angle of rotation about its centre, rather than only at the four angles listed for the square.
3. The circle has no smallest positive matching angle, because a smaller positive turn than any proposed one also makes it coincide with itself.

Q8. A figure has exactly seven equally spaced matching positions in one full turn. Calculate its smallest angle as a mixed fraction, state whether it is a whole number of degrees, and explain the division used. [3 marks]

1. The smallest angle is 360° ÷ 7 = 51 3/7°, meaning 51 and three sevenths of a degree.
2. This is not a whole number of degrees: after taking seven groups of 51 degrees from 360 degrees, three degrees remain.
3. Division by seven is appropriate because the seven matching positions are equally spaced around a complete turn. Each step therefore occupies one seventh of 360°.

Key takeaways

• Symmetry involves parts repeating in a definite pattern; a useful explanation identifies how the matching parts are related.
• A symmetry line gives complete overlap on folding, and a square has four such lines, including both diagonals.
• Reflection and rotation are different tests; the paper windmill illustrates rotational symmetry without any line of symmetry.
• Every figure matches after 360°, but rotational symmetry requires a matching turn strictly between zero and 360 degrees.
• Matching radial arms need suitable spacing; three equally spaced arms have symmetry angles of 120°, 240° and 360°.
• For finitely many equally spaced matching positions, divide a full turn by their number to find the smallest angle.
• A circle matches at every angle about its centre and has reflection symmetry along every diameter.
• In coloured patterns, check the colours as well as the outer boundary before accepting a fold or turn.

Test yourself

What makes two parts mirror halves?

They overlap completely when the figure is folded along the line separating them.

Which four lines give reflection symmetry in a square?

The central vertical and horizontal lines and both diagonals give matching folded halves.

Why is a 360° match insufficient to prove rotational symmetry?

Every figure returns after a full turn; a matching angle strictly between 0° and 360° is required.

What is the centre of rotation?

It is the fixed point about which a figure is turned when checking its matching positions.

What angles belong to a design with exactly three equally spaced matching positions?

The angles are 120°, 240° and 360°, measured about the same centre.

Five identical radial arms are equally spaced. What is the smallest matching turn?

It is 72°, found by dividing the full turn of 360° by five.

Does an unmarked circle have a smallest positive angle of symmetry?

No. It matches at every angle about its centre, so a smaller positive matching turn can always be chosen.

Why must colours be checked in a design made with tiles?

The complete coloured arrangement must match; matching the outlines alone does not establish the design's symmetry.