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NCERT Solutions · Class 6 Maths · Chapter 9

Chapter 9: Symmetry (Symmetry)

Step-by-step answers to every "Figure it Out" and in-text question of Chapter 9, Symmetry (NCERT Class 6 Maths, Ganita Prakash, 2026-27): lines of symmetry, folding and punching, paper cutting, completing symmetric figures, rotational symmetry and angles of symmetry, with diagrams. All 35 questions are answered, with the key answer highlighted.

A line of symmetry divides a figure into two mirror halves that overlap exactly when folded. A figure has rotational symmetry if it looks the same after turning it about a point by less than a full turn; each such angle is an angle of symmetry. All angles of symmetry are multiples of the smallest one, and the number of angles of symmetry (including 360°) is the order of rotational symmetry.

9.1 Line of Symmetry

Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?

Solution

Yes: the flower has 6 lines of symmetry, the rangoli has 4 and the butterfly has 1 (down its middle). The pinwheel has no line of symmetry (only rotational symmetry), and the cloud has no line of symmetry: no fold makes its two halves match.

Flower 6, rangoli 4, butterfly 1; the pinwheel and the cloud have none.

For each of the following figures, identify the line(s) of symmetry if it exists.

Solution

Figure 1 (irregular six-sided shape with a notch): no line of symmetry.
Figure 2 (kite shape): 1 line, through the two vertices where the equal sides meet.
Figure 3 (four-sided shape with one slanting side): no line of symmetry.
Figure 4 (L-shape with equal arms): 1 line, the diagonal through the corner of the L.
Figure 5 (triangle with unequal sides): no line of symmetry.

Only the kite (1 line) and the L-shape with equal arms (1 diagonal line) have lines of symmetry.

Think 1

Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have? Is the diagonal of a rectangle that is not a square a line of symmetry?

Solution

After folding vertically, horizontally and along both diagonals, there is no other way. A square has 4 lines of symmetry.

For a rectangle that is not a square, the diagonal is not a line of symmetry: when folded along a diagonal, the corners do not land on each other (the halves do not match). A rectangle has only 2 lines of symmetry, through the midpoints of opposite sides.

Square: 4 lines (2 through midpoints of sides, 2 diagonals). A rectangle's diagonal is not a line of symmetry.

Think 2

In the square ABCD, what if we reflect along the diagonal from A to C? Where do points A, B, C and D go? What if we reflect along the horizontal line of symmetry?

Solution

Along the diagonal AC: A and C stay where they are (they are on the fold), while B and D swap places.
Along the horizontal line of symmetry: each corner goes to the corner directly above or below it: A ↔ D and B ↔ C (the top and bottom corners swap).

Diagonal AC: A and C stay, B and D swap. Horizontal line: the top corners swap with the bottom corners (A ↔ D, B ↔ C).

Think 3

A sheet of paper is folded and a cut is made along the dotted line. Draw a sketch of how the paper will look when unfolded. Do you see a line of symmetry in this figure? What is it?

Solution

When unfolded, the cut-out appears on both halves, as mirror images of each other. The figure has a line of symmetry: the fold line itself.

The cut repeats as a mirror image on both halves; the fold is the line of symmetry.

In each figure, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole. How was the paper folded?

Solution

a. Two holes at the left and right edges, at the same height: folded along the vertical middle line.
b. Two holes near the top-right corner, placed on either side of the diagonal: folded along the diagonal (from the top-right to the bottom-left corner).
c. Two holes on the right edge, one near the top and one near the bottom: folded along the horizontal middle line.
d. Four holes near the four corners from a single punch: the paper was folded twice, vertically and then horizontally (or the other way round), and one hole was punched through all four layers.

a. vertical fold b. diagonal fold c. horizontal fold d. folded vertically and then horizontally, punched once through four layers.

Given the line(s) of symmetry, find the other hole(s).

Solution

Reflect each hole in the line of symmetry: draw a perpendicular from the hole to the line and continue it the same distance on the other side.

a. (diagonal from top-left to bottom-right; hole near the top-left on the left side) → the other hole is just across the diagonal, near the top-left corner on the top side.
b. (horizontal line; hole in the lower right) → the other hole is in the upper right, the same distance above the line.
c. (vertical line through the triangle's apex; hole just left of it) → the other hole is just right of the line, at the same height.
d. (slanting diameter of the circle; hole above it on the right) → the mirror hole is below the diameter, at the same distance from it.
e. (another slanting diameter; hole near the top) → the mirror hole is on the other side of the diameter, the same distance from it.

Each missing hole is the mirror image of the given hole: the same distance from the line of symmetry, on the other side, along a perpendicular to the line.

Here are some questions on paper cutting (vertical and horizontal folds). Question 4: After each of the following cuts, predict the shape of the hole when the paper is opened.

Solution

a. (vertical fold, zig-zag cut along the fold) → a symmetric zig-zag hole, like a star or a crown shape, centred on the fold line.
b. (folded paper with a V-shaped notch cut into the edge) → the notch is cut through both layers, so when opened there are two matching V-shaped notches, mirror images of each other across the fold.
c. (folded twice, a square notch cut at the folded corner) → a square hole in the middle of one side area, symmetric about both folds, as shown in the last picture.
d. (vertical fold, two rectangular notches cut on the edge) → notches on both sides, giving an "I"-shaped (dumb-bell) figure, symmetric about the fold.

The key idea: whatever is cut on the folded paper appears again as a mirror image across every fold. Make the cuts and check.

Every cut is repeated as a mirror image across the fold(s), so the holes come out symmetric about the fold lines.

Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it? a. The hole in the centre is a square (with sides parallel to the edges). b. The hole in the centre is a square standing on a corner.

Solution

a. Fold the paper in half vertically and then horizontally; the centre of the paper is now the folded corner. Fold once more along the line that halves this corner (a diagonal fold). Now make one straight cut perpendicular to one of the first two fold lines (parallel to an edge of the paper), close to the corner. When opened, the hole is a square with sides parallel to the edges.

b. Fold the paper in half vertically and then horizontally. At the folded corner, make one straight slanting cut across the corner (at 45°). When opened, the hole is a square standing on its corner (a diamond).

Check that the hole has all sides equal and all angles 90°.

a. Fold in half twice, then along the diagonal, and make one straight cut across perpendicular to that diagonal. b. Fold in half twice and make one slanting straight cut across the folded corner.

How many lines of symmetry do these shapes have? a. A square standing on a corner, and an eight-pointed star. b. A triangle with equal sides and equal angles. c. A hexagon with equal sides and equal angles.

Solution

a. The square: 4 lines. The eight-pointed star: 8 lines (4 through opposite points and 4 through the middle of opposite notches).

b. Equilateral triangle: 3 lines (from each corner to the middle of the opposite side).

c. Regular hexagon: 6 lines (3 through opposite corners and 3 through the middles of opposite sides).

a. square 4, star 8 b. 3 c. 6

Trace each figure and draw the lines of symmetry, if any.

Solution

Figure	Lines of symmetry
Three rhombuses (one on top, two below)	1 (vertical)
Three rhombuses in a row	2 (vertical and horizontal)
Four rhombuses forming a big rhombus	2 (vertical and horizontal)
Five rhombuses (two, one, two)	2 (vertical and horizontal)
Square with a smaller square (divided into 4) at its centre	4 (vertical, horizontal and the 2 diagonals)
Octagon (taller than it is wide)	2 (vertical and horizontal)
Pentagon pointing left	1 (horizontal)
Four-pointed star	4 (vertical, horizontal and the 2 diagonals)

1, 2, 2, 2 for the rhombus figures; 4 for the squares, 2 for the octagon, 1 for the pentagon and 4 for the star.

Find the lines of symmetry for the kolam.

Solution

The kolam has 2 lines of symmetry: the vertical line through its centre and the horizontal line through its centre. (The small slanting pieces in the second and fourth rows lean in mirror directions on the left and right, and the top and bottom rows mirror each other.)

2 lines: the vertical and the horizontal line through the centre.

Draw the following: a. A triangle with exactly one line of symmetry b. A triangle with exactly three lines of symmetry c. A triangle with no line of symmetry. Is it possible to draw a triangle with exactly two lines of symmetry?

Solution

a. Isosceles triangle: 1 line; b. equilateral triangle: 3 lines; c. scalene triangle: no line of symmetry

a. An isosceles triangle (two equal sides) has exactly 1 line. b. An equilateral triangle has 3. c. A scalene triangle (all sides different) has none.

Exactly two lines is not possible. If a triangle has two lines of symmetry, each line makes two sides equal, so all three sides become equal, and then the triangle (being equilateral) has three lines, not two.

a. isosceles b. equilateral c. scalene. Two lines is impossible: two lines force all sides equal, which gives three lines.

Draw the following. In each case, the figure should contain at least one curved boundary. a. A figure with exactly one line of symmetry b. A figure with exactly two lines of symmetry c. A figure with exactly four lines of symmetry

Solution

a. A semicircle, a heart shape, or an arch-shaped door: 1 line (vertical).

b. An oval (ellipse), an eye shape (two arcs meeting at points), or a rectangle with semicircles on its two short sides (a running-track shape): 2 lines.

c. A square with a semicircle bulging out from each side (or four equal circles arranged at the corners of a square, or a four-petal flower): 4 lines.

a. semicircle b. oval or running-track shape c. square with a semicircle on each side

Copy the following on squared paper. Complete them so that the blue line is a line of symmetry.

Solution

For each corner of the red figure, count how many squares it is from the blue line, and mark the matching point the same number of squares on the other side, along a line perpendicular to the blue line. Then join the new points in the same order.

For a vertical or horizontal blue line, count squares straight across (left–right or up–down).
For a slanting blue line (c and f), count diagonally across the squares (rotating the book so that the line looks vertical helps): a point that is 2 squares to the right and 2 squares down from the line becomes a point 2 left and 2 up.

Reflect each corner across the blue line, the same number of squares on the other side (diagonally for slanting lines), and join the points.

Copy the following drawings on squared paper. Complete each one so that the resulting figure has the two blue lines as lines of symmetry.

Solution

Reflect the red part in one blue line, then reflect the whole drawing (original + new part) in the other blue line. The red part appears four times, once in each of the four regions made by the two blue lines. (For the crossed diagonal lines in (a) and (b), reflect diagonally across the squares.)

Reflect the drawing in one blue line, then reflect everything in the other, so the design appears in all four regions.

Copy the following on a dot grid. For each figure, draw two more lines to make a shape that has a line of symmetry.

Solution

Choose a line of symmetry first (often the line through the middle of the figure, or the line joining its two ends), then draw the two missing sides as mirror images of the given sides. For example, for an open "V" or slanting pair of sides, the two new lines are the reflections of the given ones, making a kite or an arrowhead; for a figure with a horizontal top side and a slanting side, add the mirror of the slanting side and the closing side, making an isosceles triangle or a symmetric trapezium.

Pick a line of symmetry and draw the two new lines as the mirror images of the given lines (giving shapes such as kites, arrowheads or isosceles triangles).

9.2 Rotational Symmetry

Think 1

Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.

Solution

a. Use 5 arms, each $360° \div 5 = 72°$ from the next. Angles of symmetry: 72°, 144°, 216°, 288°, 360°.

b. Use 6 arms, $60°$ apart. Angles of symmetry: 60°, 120°, 180°, 240°, 300°, 360°.

Radial arms: 5 arms at 72° give 5 angles of symmetry; 6 arms at 60° give 6

a. 5 arms at 72°: 72°, 144°, 216°, 288°, 360° b. 6 arms at 60°: 60°, 120°, 180°, 240°, 300°, 360°

Think 2

Consider a figure with radial arms having exactly 7 angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.

Solution

$360° \div 7 = 51 \frac{3}{7} °$

It is not a whole number of degrees.

$51 \frac{3}{7} °$ (not a whole number)

Find the angles of symmetry for the given figures about the point marked: a. a plus shape b. a line with small squares at both ends on the same side c. a T-shape (vertical line with a horizontal arm, point on the arm)

Solution

a. 90°, 180°, 270°, 360° (the plus shape looks the same after every quarter turn).

b. 360° only: the small squares are on the same side, so a half turn does not bring the figure back.

c. 180°, 360°: the figure (two vertical lines joined by a horizontal one, point at the middle) looks the same after a half turn.

a. 90°, 180°, 270°, 360° b. 360° only c. 180°, 360°

Which of the following figures have more than one angle of symmetry?

Solution

Circle with two perpendicular diameters (one coloured): 180°, 360° ✓
Equilateral triangle with its centre marked: 120°, 240°, 360° ✓
Circle with three radii at 120°: 120°, 240°, 360° ✓
Four-petal shape (two coloured petals opposite each other): 180°, 360° ✓
Two crossing lines (X): 180°, 360° ✓
Five-pointed star: 72°, 144°, 216°, 288°, 360° ✓
Two half circles joined at a point on one side: only 360° ✗

All the figures except the last one (the two half circles on one side) have more than one angle of symmetry.

All except the figure made of two half circles on one side.

Give the order of rotational symmetry for each figure.

Solution

Figure	Order
Line with flags at both ends pointing opposite ways	2
Two crossing lines (X)	2
Six-pointed star	6
Three bent arms (legs)	3
Plus shape	4
Regular pentagon	5

2, 2, 6, 3, 4, 5

Think 3

In each case, the angles of symmetry are multiples of the smallest angle. Will this always happen? True or False: • Every figure will have 360 degrees as an angle of symmetry. • If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.

Solution

Yes, it always happens. If turning by the smallest angle brings the figure back, turning by it again (twice the angle), three times, and so on, also brings it back. And there can be no other angle in between, otherwise turning back by the smallest angle would give an even smaller one.

Every figure has 360° as an angle of symmetry: True (a full turn always brings a figure back to itself).
If the smallest angle is a natural number of degrees, it is a factor of 360: True (its multiples must reach 360° exactly).

Yes, always multiples of the smallest angle. Both statements are true.

9.2 Rotational Symmetry: Figure it Out

Colour the sectors of the circle (12 equal sectors) so that the figure has i) 3 angles of symmetry, ii) 4 angles of symmetry, iii) what are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?

Solution

Colour every 4th sector (3 coloured) for 3 angles of symmetry, or every 3rd sector (4 coloured) for 4

i) Colour every 4th sector (3 sectors, 120° apart): angles 120°, 240°, 360°.

ii) Colour every 3rd sector (4 sectors, 90° apart): angles 90°, 180°, 270°, 360°.

iii) The pattern must repeat after a whole number of sectors that divides 12, so the possible numbers of angles of symmetry are 1, 2, 3, 4, 6 and 12 (the factors of 12). For example, colouring alternate sectors gives 6; colouring all (or none) gives 12.

i) every 4th sector ii) every 3rd sector iii) 1, 2, 3, 4, 6 or 12 angles of symmetry (the factors of 12).

Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.

Solution

Both figures have reflection symmetry and rotational symmetry

For example: a rectangle (2 lines, angles 180° and 360°) and an equilateral triangle (3 lines, angles 120°, 240°, 360°). Others: a regular hexagon, a plus shape, a five-pointed star.

For example a rectangle and an equilateral triangle (also a regular hexagon or a plus shape).

Draw, wherever possible, a rough sketch of: a. A triangle with at least two lines of symmetry and at least two angles of symmetry. b. A triangle with only one line of symmetry but not having rotational symmetry. c. A quadrilateral with rotational symmetry but no reflection symmetry. d. A quadrilateral with reflection symmetry but not having rotational symmetry.

Solution

a. An equilateral triangle: 3 lines and 3 angles of symmetry (120°, 240°, 360°).

b. An isosceles triangle (not equilateral): 1 line, and only 360° as an angle (no rotational symmetry).

c. A parallelogram (not a rectangle or rhombus): angles of symmetry 180° and 360°, but no line of symmetry.

d. A kite (or an isosceles trapezium): 1 line of symmetry, but no rotational symmetry.

c. A parallelogram has angles of symmetry 180° and 360° but no line of symmetry; d. a kite has one line of symmetry but no rotational symmetry

a. equilateral triangle b. isosceles triangle c. parallelogram d. kite (or isosceles trapezium)

In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

Solution

The multiples of 60° up to 360°: 120°, 180°, 240°, 300°, 360°.

120°, 180°, 240°, 300°, 360°

In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?

Solution

60° must be a multiple of the smallest angle, and exactly two multiples of it lie below 60°. So 60° is the third multiple: the smallest angle is $60° \div 3 =$ 20° (the angles are 20°, 40°, 60°, …).

20°

Can we have a figure with rotational symmetry whose smallest angle of symmetry is: a. 45°? b. 17°?

Solution

a. Yes: 45° is a factor of 360° (360 ÷ 45 = 8), e.g. a figure with 8 radial arms.

b. No: 17° is not a factor of 360° (360 ÷ 17 is not a whole number), so turning by 17° again and again never comes back exactly to 360°.

a. Yes (8 × 45° = 360°) b. No (17 does not divide 360)

This is a picture of the new Parliament Building in Delhi. a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetry. How many are they? b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.

Solution

The outer boundary is a hexagon with three long sides and three short sides, placed alternately (a triangle with its corners cut off).

a. Yes, 3 lines of symmetry, each through the middle of a long side and the middle of the opposite short side.

b. Yes, it has rotational symmetry of order 3: angles 120°, 240° and 360°.

a. Yes, 3 lines b. Yes: 120°, 240°, 360°

How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?

Solution

A regular polygon with $n$ sides has $n$ lines of symmetry: triangle 3, square 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10. The sequence is 3, 4, 5, 6, 7, 8, 9, 10: the counting numbers starting from 3.

3, 4, 5, …, 10: the counting numbers from 3.

How many angles of symmetry do the shapes in the first shape sequence (Regular Polygons) have? What number sequence do you get?

Solution

A regular polygon with $n$ sides has $n$ angles of symmetry (multiples of $360° \div n$ ): again 3, 4, 5, 6, 7, 8, 9, 10, the counting numbers starting from 3.

3, 4, 5, …, 10: the counting numbers from 3.

How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?

Solution

The first shape (an equilateral triangle) has 3 lines and 3 angles of symmetry. Every later shape in the sequence (the snowflakes) has 6 lines and 6 angles of symmetry (60°, 120°, …, 360°). So: lines 3, 6, 6, 6, …; angles 3, 6, 6, 6, …

Lines: 3, 6, 6, 6, …; angles of symmetry: 3, 6, 6, 6, …

How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?

Solution

It has 24 equally spaced spokes, so it has 24 lines of symmetry and 24 angles of symmetry (multiples of 15°: 15°, 30°, …, 360°).

24 lines and 24 angles of symmetry (each 15° apart).

Tiles

Playing with Tiles: a. Use the colour tiles to complete the figure so that it has exactly 2 lines of symmetry. b. Use 16 such tiles to make figures that have exactly 1 line of symmetry and exactly 2 lines of symmetry. c. Use these tiles in making creative symmetric designs.

Solution

Each tile is a square split by a diagonal into two colours.

Exactly 2 lines: arrange 4 × 4 tiles so that the left half is the mirror image of the right half and the top half is the mirror image of the bottom half (place each tile's colours as reflections across the middle vertical and horizontal lines). Check that the diagonals are not lines of symmetry.
Exactly 1 line: make the left half the mirror image of the right half only, and make the top and bottom halves different.

Experiment and check each design by imagining folding it along the lines.

Mirror the tiles across both middle lines for 2 lines of symmetry, or across one middle line only for 1 line.

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