NCERT Solutions for Class 6 Maths Chapter 9 Symmetry – step-by-step answers to every exercise and in-text question of the chapter, as per the latest NCERT syllabus for 2027. Read them online or download the free question-answer PDF.
Chapter 9 of Ganita Prakash is about symmetry — parts of a figure that repeat in a definite pattern. It covers lines of symmetry found by folding, punching holes and cutting paper, reflection, and rotational symmetry: angles of symmetry, the centre of rotation and figures with radial arms. It ends with the circle, regular polygons, the Koch snowflake, the Ashoka Chakra, a tile activity and a grid game.
NCERT Solutions for Class 6 Maths Chapter 9 Symmetry PDF
All the answers are on this page below. The PDF also explains every answer – why it is right and how to check it – so download it for revision.
| Chapter | Download | |
|---|---|---|
| Solutions PDF (English medium) | Symmetry | View PDF |
| Solutions PDF (Hindi medium) | सममिति | View PDF |
NCERT Solutions for Class 6 Maths Chapter 9 Symmetry – Questions and Answers
All 59 questions of the chapter (textbook pages 217 – 241) are answered below, in the order of the book – in-text questions, activities and the end-of-chapter exercises. Each answer is written the way you would write it in the exam.
In-text Questions — Pages 217 & 218Chapter opening — Flower, Butterfly, Rangoli, Pinwheel
What appeals to us in the butterfly is that its left half and right half are exactly alike.
The red petals repeat. There are 4 red petals, and a quarter turn (90°) about the centre brings each red petal onto the next one. The ring of 8 green petals repeats in the same way.
The centre of the pinwheel is a regular hexagon. A rhombus — two equilateral triangles joined along a side — is stuck to each of its six sides.
How the shapes move: going round the hexagon, each rhombus is the one before it turned through 60° about the centre. All six lean the same way, like the blades of a fan.
The other pictures: the flower’s petals repeat round its centre, the butterfly’s two wings are mirror halves, and the rangoli’s petals repeat at every quarter turn.
Taj Mahal: seen from the front, it has one vertical line of symmetry through the middle of the dome and the main arch. The left and right halves — arches, small domes and minarets — are mirror halves.
Gopuram: it also has one vertical line of symmetry, and the same row of carved figures repeats storey after storey, a little smaller each time.
Figure it Out — Page 219Section 9.1 Line of Symmetry
Yes — three of the four figures can be folded into mirror halves; the pinwheel and the cloud cannot.
| Figure | Lines of symmetry | Where they are |
|---|---|---|
| Flower (6 petals) | 6 | 3 through opposite petals and 3 between petals |
| Rangoli | 2 | the vertical and the horizontal line through the centre |
| Butterfly | 1 | the vertical line along its body |
| Pinwheel | 0 | every blade leans the same way, so no fold works |
| Cloud | 0 | nothing in it repeats at all |
Three figures have exactly one line of symmetry (dashed red lines below); two have none.
- (i) arrow shape — 1 line, through the V-notch and the opposite corner
- (ii) kite — 1 line, the vertical line joining the top and bottom corners
- (iii) four-sided figure with unequal upright sides — no line
- (iv) L-shape — 1 line, the slanting line through the outer and the inner corner
- (v) triangle with three unequal sides — no line
In-text Questions — Pages 221 & 222Figures with more than one line of symmetry; Reflection
No, there is no fifth way. A square has exactly 4 lines of symmetry.
- Fold 1 — vertical fold (left half onto right half).
- Fold 2 — horizontal fold (top half onto bottom half).
- Fold 3 — along one diagonal.
- Fold 4 — along the other diagonal.
Yes. All three figures on page 221 have more than one line of symmetry.
| Figure | Lines of symmetry | Angles of symmetry |
|---|---|---|
| The red snowflake (Koch snowflake) | 6 | 60°, 120°, 180°, 240°, 300°, 360° |
| The yellow flower with 6 petals | 6 | 60°, 120°, 180°, 240°, 300°, 360° |
| The green knot design (like a looped square) | 4 | 90°, 180°, 270°, 360° |
No. The diagonal of a rectangle is not a line of symmetry.
On folding a rectangular sheet along its diagonal, the two parts do not overlap — two corners stick out.
The square is labelled A (top left), B (top right), C (bottom right) and D (bottom left).
Reflection in the diagonal AC:
Reflection in the horizontal line of symmetry:
• What do you see?
• Is the resulting figure symmetric?
• If yes, where is the line of symmetry?
• Is there any other line along which it can be folded to produce two identical parts?
- What you see: two identical blots, one on each side of the crease.
- Is it symmetric? Yes — the two halves are mirror halves.
- Line of symmetry: the crease (the fold line).
- Any other line? Usually no — a random blot has only this one line.
In-text Questions — Page 223Paper Folding and Cutting
When the paper is opened, there is one hole in the middle of the sheet: the cut piece together with its mirror image in the fold — a six-sided hole, pointed at the top and the bottom, with straight upright sides.
Yes, the opened sheet has a line of symmetry — the fold line (crease). Everything on one side of it is the mirror image of the other side, hole included.
The whole printed design has 2 lines of symmetry: the horizontal line through the centres of the three stars, and the vertical line through the middle star.
Each eight-pointed star also has 8 lines of symmetry and each small rhombus has 2. When a strip is folded like a fan and cut, every crease becomes a line of symmetry of the opened strip.
Figure it Out — Pages 223 to 230Punching Game, Paper Cutting and Lines of Symmetry
- (a) along the vertical line through the middle
- (b) along the diagonal from the bottom-left corner to the top-right corner
- (c) along the horizontal line through the middle
- (d) the paper was folded twice — in half vertically and then in half horizontally (or the other way round) — so one punch went through 4 layers and made 4 holes
Mark each new hole at the same distance on the other side of the line of symmetry, on the line through the given hole at right angles to it. Each figure gets one new hole:
This part only explains the two pictures that will be used in question 4. Learn to read them:
| Picture | What it means | Where the crease is | What happens to a cut |
|---|---|---|---|
| Vertical Fold | the right half of the sheet is turned over onto the left half (or the left onto the right) | a vertical line through the middle | every cut is copied to the left and right — the opened sheet has a vertical line of symmetry |
| Horizontal Fold | the top half is turned down onto the bottom half (or the bottom up) | a horizontal line through the middle | every cut is copied above and below — the opened sheet has a horizontal line of symmetry |
In every case the answer is found the same way: take the piece cut away and add its mirror image in the crease.
- (a) one hole on the crease, narrowest at the crease, with wavy sides (like a bow tie)
- (b) a band-shaped hole across the middle with a V-notch at each end
- (c) two holes on the vertical crease, and two notches in both the top and the bottom edge
- (d) a rectangular hole in the middle and a notch in each side edge
a. The hole in the centre is a square.
b. The hole in the centre is a square.
(a) Fold the sheet along one diagonal, then along the other. The centre is now the right-angled corner of a small triangle. One straight cut across that corner, cutting equal lengths along both folds, opens into an upright square.
(b) Fold the sheet in half, then in half again (a vertical and a horizontal fold). One slanting cut across the closed corner, cutting equal lengths along both folds, opens into a square standing on a corner.
a.
- a. square standing on a corner: 4 lines; eight-pointed star: 8 lines
- b. equilateral triangle: 3 lines
- c. regular hexagon: 6 lines
The four rhombus (diamond) patterns:
| Figure | Lines of symmetry | Where |
|---|---|---|
| Three rhombuses — one on top, two below | 1 | the vertical line through the top rhombus |
| Three rhombuses in a row | 2 | the vertical line through the middle rhombus and the horizontal line through all three centres |
| Four rhombuses making one big rhombus | 2 | the vertical and the horizontal line through the centre |
| Four rhombuses in two slanting pairs | 2 | the vertical and the horizontal line through the centre |
The four figures drawn on squared paper:
The kolam has 6 lines of symmetry, all passing through the centre star:
- 3 lines that pass through pairs of opposite outer stars;
- 3 lines that pass between the stars, through the rhombuses.
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?
- a. an isosceles triangle (two sides equal), e.g. 6 cm, 6 cm, 4 cm — 1 line
- b. an equilateral triangle, e.g. all sides 5 cm — 3 lines
- c. a scalene triangle (all sides different), e.g. 4 cm, 6 cm, 7 cm — no line
Is a triangle with exactly two lines of symmetry possible? No, it is impossible.
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.
- a. a semicircle — exactly 1 line
- b. an oval (ellipse) — exactly 2 lines
- c. a flower with four equal round petals (or a square with a semicircle on each side) — exactly 4 lines
For every corner of the red drawing, count the squares to the blue line and mark a point the same number of squares on the other side; then join the new points in the same order. The completed figures:
Do the reflection twice:
- Reflect the given red piece in the first blue line.
- Now reflect both pieces (the given one and the new one) in the second blue line.
This gives 4 copies of the red piece, one in each of the four parts made by the blue lines. The completed figures:
Many answers are possible. One answer for each figure — reflect the given lines in a line through the dots, and close the shape with two new lines:
In-text Questions — Pages 230 & 231Section 9.2 Rotational Symmetry
No. The paper windmill has four identical blades placed at equal distances round the centre.
Only when the turn reaches 90° does each blade land exactly where the next blade was, and the windmill looks untouched.
Yes, many. Any figure built by repeating one piece four times around a centre has exactly 4 angles of symmetry — 90°, 180°, 270°, 360°.
For example: a square; a plus sign with four equal arms (like the figure on page 232); a four-petal flower; the four-pointed star of page 227; a fan with 4 equal blades.
A square has 4 angles of symmetry.
- Rotation needed: a quarter turn, 90° — A goes to B’s place, B to C’s, C to D’s and D to A’s.
- Centre of rotation: the centre of the square, where its diagonals cross.
- Angles of symmetry: 90°, 180°, 270° and 360°.
Example (page 232): Find the angles of symmetry of the following strip. The strip is rotated clockwise about its centre; a rotation of 180° gives the second figure. Does this overlap with the original figure? No. Why?
Because the strip is not alike at the top and the bottom: its bottom side is longer than its top side. After a half turn the long side is at the top, so the turned strip does not cover the original one.
Only a full turn (360°) brings it back. So 360° is its only angle of symmetry — the strip has no rotational symmetry.
In-text Questions — Pages 232 to 235Rotational Symmetry of Figures with Radial Arms
The four arms are identical and the angle between neighbouring arms is 90°, so each arm can take the place of the next one.
No — not if all four arms are to be kept. With 4 arms the only way to have 4 angles of symmetry is to keep them at 90° to each other.
Make the arms unequal in pairs: keep the two opposite arms alike, and make the other two opposite arms different.
Cut out a tracing and turn it slowly over the printed figure. You will find that no partial turn ever matches.
So this figure does not have rotational symmetry, even though it has three arms.
Yes — change the angles between the arms so that all three are equal.
∠A + ∠B + ∠C = 360° (one full turn)
3 × ∠A = 360°
∠A = 360° ÷ 3 = 120°
With 120° between the adjacent dotted lines, the figure has 3 angles of symmetry:
a) exactly 5 angles of symmetry,
b) 6 angles of symmetry?
Also find the angles of symmetry in each case. Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?
a) Exactly 5 angles of symmetry — draw 5 equal arms.
Angles of symmetry = 72°, 144°, 216°, 288°, 360°
b) Exactly 6 angles of symmetry — draw 6 equal arms.
Angles of symmetry = 60°, 120°, 180°, 240°, 300°, 360°
Divide 360 by 7: 7 × 51 = 357, and 360 − 357 = 3.
No, it is not a whole number of degrees. The smallest angle of symmetry is 513⁄7° (about 51.43°).
Figure it Out — Pages 235 & 236Section 9.2 Rotational Symmetry
(a) The plus-shaped figure (two equal bars crossing at right angles):
(b) The bracket-shaped figure (a segment with a small square at each end, both on the same side):
(c) The H-shaped figure:
Five of the seven figures have more than one angle of symmetry.
| Figure | More than one angle? | Angles of symmetry |
|---|---|---|
| Circle with two perpendicular diameters | Yes | 90°, 180°, 270°, 360° |
| Triangle pointing downwards (isosceles, not equilateral) | No | 360° only |
| Circle divided into 3 equal sectors | Yes | 120°, 240°, 360° |
| Pinwheel with 4 curved blades | Yes | 90°, 180°, 270°, 360° |
| The cross “X” of two equal straight lines at right angles | Yes | 90°, 180°, 270°, 360° |
| Five-pointed star | Yes | 72°, 144°, 216°, 288°, 360° |
| Segment with two semicircles on the same side (like a “B”) | No | 360° only |
The order of rotational symmetry is the number of angles of symmetry — that is, how many times the figure fits onto itself in one full turn.
| Figure | Order | Angles of symmetry |
|---|---|---|
| Slanting segment with an arrow-flag at each end (turned opposite ways) | 2 | 180°, 360° |
| Two crossing lines with an arc drawn between the upper arms | 1 | 360° only |
| Six-pointed star | 6 | 60°, 120°, 180°, 240°, 300°, 360° |
| The three bent arms with fists (a triskelion) | 3 | 120°, 240°, 360° |
| Plus sign made of five squares | 4 | 90°, 180°, 270°, 360° |
| Regular pentagon | 5 | 72°, 144°, 216°, 288°, 360° |
In-text Questions — Pages 236 & 237Multiples of the smallest angle; True or False; Symmetries of a circle
Yes, it always happens. Every angle of symmetry of a figure is a multiple of its smallest angle of symmetry.
Exactly 3 angles: 120°, 240°, 360° → multiples of 120
Exactly 4 angles: 90°, 180°, 270°, 360° → multiples of 90
True. A full turn of 360° brings every point of any figure back to exactly where it started.
True. 360° is always an angle of symmetry, and every angle of symmetry is a multiple of the smallest one, s. So 360 = n × s for some whole number n — that is, s is a factor of 360.
On turning, the circle coincides with itself whatever the angle of turn — for a circle, every angle is an angle of symmetry.
Yes, that diameter is a line of symmetry — in fact every diameter is: folding along it, the two semicircles match exactly.
Objects around us with rotational symmetry: a ceiling fan with 3 equal blades, a flower, a wheel with equal spokes, a bangle, a plate.
Figure it Out — Pages 238 & 239Symmetries of a Circle; Regular Polygons; Ashoka Chakra
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?
Each of the 12 sectors is 360° ÷ 12 = 30°.
- (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) possible numbers of angles of symmetry: 1, 2, 3, 4, 6 and 12 — the factors of 12
Any regular polygon or any regular star works. Two easy ones:
- Equilateral triangle — 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
- Regular hexagon — 6 lines of symmetry and 6 angles of symmetry (60°, 120°, 180°, 240°, 300°, 360°).
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.
- a. an equilateral triangle — 3 lines and 3 angles of symmetry (120°, 240°, 360°)
- b. an isosceles triangle — 1 line, no rotational symmetry
- c. a parallelogram that is not a rectangle or a rhombus — no line, but a half turn (180°) brings it back
- d. a kite — 1 line, no rotational symmetry
Every angle of symmetry is a multiple of the smallest one, and 360° is always an angle of symmetry. So keep adding 60° until you reach 360°.
The other angles of symmetry are 120°, 180°, 240°, 300° and 360°.
Let the smallest angle of symmetry be s. All angles of symmetry are the multiples s, 2s, 3s, …
The figure has exactly two angles of symmetry smaller than 60°; these must be s and 2s. And since 60° itself is an angle of symmetry, the next multiple after 2s must be 60°:
s = 60° ÷ 3 = 20°
The smallest angle of symmetry is 20°.
a. 45°?
b. 17°?
The test is simple: the smallest angle of symmetry must divide 360° exactly.
a. 45°? Yes.
b. 17°? No.
a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they?
b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.
a. Yes, it has reflection symmetry — there are 3 lines of symmetry. Each one passes through the middle of a side and the opposite cut corner, and all three meet at the centre of the building.
b. Yes, it has rotational symmetry of order 3 about that centre.
Angles of rotational symmetry = 120°, 240°, 360°
| Regular polygon | Triangle | Quadrilateral | Pentagon | Hexagon | Heptagon | Octagon | Nonagon | Decagon |
|---|---|---|---|---|---|---|---|---|
| Sides | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Lines of symmetry | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
The number sequence is 3, 4, 5, 6, 7, 8, 9, 10, … — the counting numbers starting from 3. It is the very same sequence as the number of sides.
A regular polygon of n sides has exactly n angles of symmetry, because turning it by one “side” brings each corner onto the next corner.
| Regular polygon | Sides | Angles of symmetry | Smallest angle = 360° ÷ n |
|---|---|---|---|
| Triangle | 3 | 3 | 120°, 240°, 360° |
| Quadrilateral (square) | 4 | 4 | 90°, 180°, 270°, 360° |
| Pentagon | 5 | 5 | 72°, 144°, …, 360° |
| Hexagon | 6 | 6 | 60°, 120°, …, 360° |
| Heptagon | 7 | 7 | 513⁄7°, …, 360° |
| Octagon | 8 | 8 | 45°, 90°, …, 360° |
| Nonagon | 9 | 9 | 40°, 80°, …, 360° |
| Decagon | 10 | 10 | 36°, 72°, …, 360° |
The number sequence is again 3, 4, 5, 6, 7, 8, 9, 10, … — the counting numbers from 3, exactly the same sequence as in question 8.
| Shape in the sequence | 1st | 2nd | 3rd | 4th | 5th |
|---|---|---|---|---|---|
| Lines of symmetry | 3 | 6 | 6 | 6 | 6 |
| Angles of symmetry | 3 | 6 | 6 | 6 | 6 |
So both sequences are 3, 6, 6, 6, 6, …
The Ashoka Chakra at the centre of our national flag has 24 spokes, equally spaced around the hub.
Lines of symmetry = 24
Angles of symmetry = 24
Smallest angle of rotation = 360° ÷ 24 = 15°
Playing with Tiles — Pages 239 & 240Playing with Tiles
A completion with exactly 2 lines of symmetry: put the given quarter, turned upside down, in the top-right corner, and fill the bottom half so that the two diagonals of the big square are mirror lines (right-hand picture).
1 line of symmetry
2 lines of symmetry
sixteen tiles make a 4 × 4 square.
Exactly 1 line: arrange the left half freely and fill the right half with mirror images — keeping the top and the bottom different. Exactly 2 lines: arrange one quarter and mirror it across both middle lines — keeping the quarter not symmetric about its own diagonal.
(an activity — any correct design is fine): a rangoli square with 4 lines of symmetry; a pinwheel of four quarters, each turned by 90°, which has rotational symmetry but no line of symmetry; a border strip that repeats one motif with a mirror at every join.
Game — Page 241The 6 by 6 grid game
The winning idea is symmetry itself — copy your opponent.
Strategy for the second player
- Look at the centre of the 6 × 6 grid (the point where the middle horizontal and middle vertical lines cross).
- Whatever your opponent covers, you cover the pair of squares that is its half-turn (180°) image about that centre.
Chapter 9 of 10 – Maths
- Previous: Chapter 8 Playing with Constructions
- Next: Chapter 10 the Other Side of Zero
- All chapters: NCERT Solutions Class 6 Maths
Also for This Chapter
NCERT Solutions Class 6 Maths Chapter 9 Symmetry – Overview
| Class | Class 6 |
| Subject | Maths |
| Chapter | 9 – Symmetry |
| Textbook pages | 217 – 241 |
| Questions solved | 59 |
| Session | 2027 |
| Format | Full answers on this page + PDF |
NCERT Solutions Class 6 Maths Chapter 9 – FAQs
Where can I download NCERT Solutions Class 6 Maths Chapter 9 Symmetry PDF?
Read every answer on this page, or download the free PDF from the table at the top.
Is this for the 2027 session?
Yes, it follows the latest NCERT Class 6 Maths book for 2027.
How many questions are solved in NCERT Class 6 Maths Chapter 9?
All 59 questions of the chapter (textbook pages 217 – 241) – the in-text questions, activities and the end-of-chapter exercises.
How should I use these solutions?
Read the chapter in the NCERT book first, attempt the questions yourself, then check your answers here.