NCERT Solutions Class 6 Maths Chapter 9 Symmetry

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 6 · M AT H S

NCERT Solutions

Chapter 9: Symmetry

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

217 – 241 12 58 English

Solutions, notes, sample papers & more at 88 pages

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 9: Symmetry
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 9 Symmetry from the NCERT textbook
Ganita Prakash. Every Figure it Out question and every in-text question from pages 217 to 241 is solved with
diagrams — lines of symmetry by paper folding, punching and cutting, reflection, rotational symmetry, angles
of symmetry, order of rotational symmetry, regular polygons, the Koch snowflake and the Ashoka Chakra.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 217 – 241

SECTIONS QUESTIONS

12 58

MEDIUM

English

In-text Questions — Pages 217 & 218
Chapter opening — Flower, Butterfly, Rangoli, Pinwheel

MATH TALK

Q1 The flower looks the same from many different angles. What about the butterfly?
No doubt, the colours are very attractive. But what else about the butterfly appeals
to you?

ANSWER

What appeals to us in the butterfly is that its left half and right half are exactly alike.

Page 1 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

line of symmetry
Fold the butterfly along its body and the two wings cover each other exactly.

If we fold the picture along the line through the body, the left wing falls exactly on the right
wing — the same spots, the same shape, the same size. These are mirror halves.

Why it is different from the flower: the flower repeats when you turn it (rotation),
while the butterfly repeats when you flip it (reflection). The butterfly has exactly one
line of symmetry, and no rotational symmetry.

Page 2 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q2 In these pictures, it appears that some parts of the figure are repeated and these
repetitions seem to occur in a definite pattern. Can you see what repeats in the
beautiful rangoli figure?

The rangoli from page 217.

ANSWER

Yes. In the rangoli, one quarter of the design repeats four times around the centre.

The red petals, the green leaves and the small blue drops all come back onto themselves
when the rangoli is turned by 90° about its centre.
Turning again by 90° (that is 180°), again (270°) and once more (360°) also brings it back.

Angles of symmetry of the rangoli = 90°, 180°, 270°, 360°

Page 3 of 88

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Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

The rangoli also has 4 lines of symmetry — one vertical, one horizontal and the two slanting
co m
lines through the centre.
e m.
m l as
.co a g
s em
Try This: Draw only one quarter of a rangoli on a square of paper. Fold the paper
a
gl cut, and open it — the full rangoli appears by itself.
atwice,

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 4 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q3 What about the pinwheel? Can you spot which pattern is repeating? Hint: Look at
the hexagon first. Now, can you say what figure repeats along each side of the
hexagon? What is the shape of the figure that is stuck to each side? Do you
recognise it? How do these shapes move as you move along the boundary of the
hexagon?

The pinwheel from page 217.

ANSWER

Start from the middle. The centre of the pinwheel is a regular hexagon (six equal sides), and a
rhombus — made of two equilateral triangles joined along a side — is stuck to each of its six
sides.

Page 5 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

A regular hexagon with a rhombus (two equilateral triangles) stuck to each of its six sides, every
rhombus tilted the same way round.

How the shapes move: as you walk round the boundary of the hexagon, each rhombus is the
previous one turned by 60° about the centre. They all lean the same way, like the blades of a
fan.

Angles of symmetry of the pinwheel = 60°, 120°, 180°, 240°, 300°, 360° (6 angles)

Page 6 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Important: because every blade leans the same way, a mirror would tip them the
other way. So the pinwheel has no line of symmetry at all, yet it is beautifully
symmetric — this is rotational symmetry, studied in Section 9.2.

Q4 What are the symmetries that you see in these beautiful structures? (Taj Mahal and
Gopuram)

ANSWER

Taj Mahal. Looking at it from the front:

There is one vertical line of symmetry right through the middle of the dome and the main
archway. The left half and the right half — arches, small domes, and the four minarets — are
exact mirror halves.
The water channel in front adds a second, horizontal mirror: the reflection in the water is an
upside-down copy of the building.
Seen from directly above, the whole monument even has 4 lines of symmetry and turns
onto itself every 90°.

Gopuram (temple gateway tower).

It also has one vertical line of symmetry — the left and right sides of every storey match
exactly.
Going up, the same row of carved figures repeats storey after storey, each one a little smaller.
This is a repeating pattern (translation), which is why the tower looks so orderly.

Math Talk: Look at photographs of the Qutub Minar, the Konark wheel, the
Charminar and a temple kolam. For each one ask two questions — “Can I fold it?”
and “Can I turn it?”

Figure it Out — Page 219

Page 7 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Section 9.1 Line of Symmetry

Q1 Do you see any line of symmetry in the figures at the start of the chapter? What
about in the picture of the cloud?

Flower Butterfly Rangoli Pinwheel

The four pictures at the start of the chapter, page 217. The cloud on page 218 is a
photograph of an ordinary fluffy white cloud in a blue sky — described here; see the
textbook for the photograph.

ANSWER

Yes — three of the four figures can be folded into mirror halves; the pinwheel and the cloud
cannot.

FIGURE LINES OF WHERE THEY ARE
SYMMETRY

Flower (6 6 one along each petal and one between every two petals
petals)

Rangoli 4 vertical, horizontal and the two slanting lines 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

Note the difference: the pinwheel has no line of symmetry but it does look the
same after a 60° turn. The cloud has neither — it is not symmetrical in any way.

Page 8 of 88

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Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
For each of the following figures, identify the line(s) of symmetry if it exists.
e
Q2

m l as
m .co a g
l a se
ag

co m
m .
e(iii) ag
(i) (ii)
g l as (iv) (v)

a
The five figures given on page 219.

co m
em.
m l as
.co a g
ANSWER

a s em
l are the lines of symmetry.
Three of the five figures have exactly one line of symmetry; two have none. The dashed red lines

agbelow
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a (i) 1 line
(ii) 1 line (iii) none
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co g l
(v)anone
m
ase
(iv) 1 line

agl c
.
The five figures of page 219 with their lines of symmetry drawn as dashed lines.

s e m
m a
m . co
(i) The arrow-shaped figure — 1 line of symmetry: the line through the V-shaped notch
e agl
l as
and the opposite corner. Folding along it makes the upper point fall on the lower point.
g
a

co m
m .
m ase
.co


a g l Page 9 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

(ii) The kite — 1 line of symmetry: the vertical line joining the top corner to the bottom
corner. (Its other diagonal is not a line of symmetry, because the two parts are of different
sizes.)
(iii) The four-sided figure with two vertical sides of different lengths — no line of
symmetry. No fold makes the short side fall on the long side.
(iv) The L-shape — 1 line of symmetry: the slanting diagonal from the outer corner to the
inside corner. Both arms have the same length and the same thickness, so they swap over
exactly.
(v) The thin triangle — no line of symmetry. All its three sides are of different lengths (a
scalene triangle).

Check it yourself: trace each figure on paper, cut it out and try folding it in every
way you can. A fold works only when the two parts cover each other completely —
no part sticking out.

In-text Questions — Pages 221 & 222

Page 10 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Figures with more than one line of symmetry; Reflection

TRY THIS

Q1 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?

Fold 1

Fold 2

Fold 3 Fold 4
A square piece of paper and the four folds shown on page 220.

ANSWER

No, there is no fifth way. A square has exactly 4 lines of symmetry.

Page 11 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

The four folds of a square: vertical, horizontal and the two diagonals.

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.

Why no more: a line of symmetry of a square must pass through its centre and
must send corners to corners. There are only 4 such lines — two through the
midpoints of opposite sides, and two through opposite corners.

Page 12 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q2 Thus, figures can have multiple lines of symmetry. The figures below also have
multiple lines of symmetry. Can you find them all?

The three figures on page 221 — a snowflake, a six-petal flower and a looped-square
design.

ANSWER

Yes. All three figures on page 221 have more than one line of symmetry.

FIGURE LINES OF ANGLES OF SYMMETRY
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 4 90°, 180°, 270°, 360°
square)

Page 13 of 88

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Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
e m.
m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
.
A six-pointed figure has 6 lines of symmetry — 3 through opposite points and 3 through opposite
m a
ase
notches. The snowflake and the 6-petal flower work exactly like this.

agl
m
How to count quickly: if a design is built by repeating one piece n times around a
. co
m
centre and the piece itself is not tilted, the design has n lines of symmetry and n

m as e
.co
angles of symmetry.
a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q3 We saw that the diagonal of a square is also a line of symmetry. Let us take a
rectangle that is not a square. Is its diagonal a line of symmetry? First, see the
rectangle and answer this question. Then, take a rectangular piece of paper and
check if the two parts overlap by folding it along its diagonal. What do you observe?

ANSWER

No. The diagonal of a rectangle is not a line of symmetry.

diagonal ✗

the two real lines of symmetry ✓
A rectangle has only 2 lines of symmetry (green). The diagonal (red) is not one of them.

When you fold a rectangular sheet along its diagonal, the two triangles do not cover each
other — one long edge sticks out on one side and one short edge sticks out on the other.

Why it happens: a fold along the diagonal would have to send the long side (say 8
cm) onto the short side (say 5 cm). Folding cannot change a length, so 8 cm can
never land on 5 cm. In a square all four sides are equal, which is exactly why its
diagonals do work.

So a rectangle that is not a square has exactly 2 lines of symmetry: the line through the
midpoints of the two long sides and the line through the midpoints of the two short sides.

Page 15 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q4 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?

A B

D C
The square ABCD with its vertical line of symmetry, page 222.

ANSWER

The square is labelled A (top left), B (top right), C (bottom right) and D (bottom left).

Page 16 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

A B

D C
The square ABCD with the vertical line (red), the horizontal line (green) and the diagonal AC (purple).

Reflection in the diagonal AC:

A stays at A C stays at C B goes to D D goes to B

A and C lie on the mirror line, so they do not move. B and D are on opposite sides of it, at equal
distances, so they simply change places.
Reflection in the horizontal line of symmetry:

A↔D B↔C

The top edge falls on the bottom edge, so A takes the position of D, D takes the position of A, B
takes the position of C and C takes the position of B.

Page 17 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

MIRROR LINE A GOES TO B GOES TO C GOES TO D GOES TO

Vertical line B A D C

Horizontal line D C B A

Diagonal AC A D C B

Diagonal BD C B A D

Q5 Ink Blot Devils: Take a piece of paper. Fold it in half. Open the paper and spill a few
drops of ink (or paint) on one half. Now press the halves together and then open the
paper again. • 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?

ANSWER

What you see: the ink spreads and prints a copy of itself on the other half. You now see two
identical blots, one on each side of the crease — often they look like a butterfly, a bat or a
devil's face. That is why it is called an ink blot devil.
Is it symmetric? Yes. Whatever the blot's shape, the two halves are mirror halves of each
other.
Where is the line of symmetry? Exactly along the crease — the line you folded the paper
on. The crease is the mirror.
Any other line? Usually no. A random blot has just this one line of symmetry. You get a
second line only if the ink drops themselves were placed symmetrically about a line at right
angles to the crease.

Why it works: pressing the halves together carries every ink point to the point
directly opposite it, at the same distance from the crease, on the other side. That is
exactly what a reflection does.

Try This: fold the paper twice (once vertically and once horizontally), drop ink in the
quarter, press, and open. Now the pattern has 2 lines of symmetry and looks the
same after a half turn.

In-text Questions — Page 223

Page 18 of 88

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Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

Paper Folding and Cutting
co m
e m.
m as
TRY THIS

.co a g l
s m two figures, a sheet of paper is folded and a cut is made along the dotted
ethese
gl aIn
a
Q1
line shown. Draw a sketch of how the paper will look when unfolded. Do you see a
line of symmetry in this figure? What is it?

co m
em . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
c o agl
fold fold
.
malong the dotted line
s e
cut

ag la
The two pictures on page 223 — a sheet folded along its left edge, with the dotted line
along which the cut is made.

co m
m .
o m l a se
.c a g
m
se cut is made starting from the folded edge, so the piece that falls out is only half of the
ANSWER

g l a
a
The
hole. When the paper is opened, that half is joined by its mirror image and a complete hole

se m
com l a
appears in the middle of the sheet.
. a g
m
ase
agl

co m
m .
m as e
.co a g l
se m →
g l a
a c
m .
m a s e
e m . co agl
g l as
fold a line of symmetry

com
m .
m ase
.co


a g l Page 19 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Left: the folded sheet with the dotted cut. Right: the opened sheet — the hole is the cut shape
together with its mirror image.

Yes, the opened figure has a line of symmetry — it is exactly the crease (the fold line).
Everything on the right of the crease is the mirror image of everything on its left, hole included.

Tip: the shape of the hole is always “the cut piece + its mirror image”. So if you cut
half a heart at the fold, you open out a full heart; half a star gives a full star.

Q2 Use thin rectangular coloured paper. Fold it several times and create some intricate
patterns by cutting the paper, like the one shown here. Identify the lines of
symmetry in the repeating design. Use such decorative paper cut-outs for festive
occasions.

The repeating cut-paper design shown on page 223.

ANSWER

Fold a long strip of thin paper like a fan (accordion folds), cut a small design through all the
layers at once, and open it out. You get the same design repeated again and again along the
strip.

Each cut passes through every layer, so one snip makes as many copies as there are layers.
Four folds give 16 layers and 16 copies!
Every crease is a line of symmetry of the finished strip — the design on one side of a crease
is the mirror image of the design on the other side.

Page 20 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

In the printed design each blue star also has its own lines of symmetry: an eight-pointed star
has 8, and each small diamond has 2.

Two kinds of pattern in one strip: the whole strip repeats by sliding (translation)
along its length, and each repeat is a mirror of its neighbour. That is why the border
looks so neat.

Try This: make such a paper toran for Diwali or a birthday. Fold, cut a half-flower on
the crease, open — a whole row of flowers appears.

Figure it Out — Pages 223 to 230
Punching Game, Paper Cutting and Lines of Symmetry

TRY THIS

Q1 In each of the following figures, 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?

a. b. c. d.

The four unfolded sheets (a) to (d) on page 224.

ANSWER

Two holes made by one punch are always mirror images of each other in the fold line. So the
fold is the line that is exactly halfway between them and at right angles to the line joining them.

Page 21 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

(a) vertical (b) diagonal (c) horizontal (d) both folds

The fold line is the mirror line of the holes. In (d) two folds were used, so one punch gave four holes.

(a) The two holes are side by side at the same height. The paper was folded along the
vertical line through the middle.
(b) The two holes lie slanting, one above the other towards the top corner. The paper was
folded along the diagonal of the square (from the bottom-left corner to the top-right
corner).
(c) The two holes are one above the other near the right edge. The paper was folded along
the horizontal line through the middle.
(d) There are four holes, one at each corner, from a single punch. So the paper was folded
twice — first in half vertically and then in half horizontally (or the other way round). The
punch then went through 4 layers.

1 fold → 2 layers → 2 holes

2 folds → 4 layers → 4 holes

3 folds → 8 layers → 8 holes

Page 22 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q2 Given the line(s) of symmetry, find the other hole(s):

a. b. c. d. e.

The five sheets printed with this question on page 224. In each one the dashed line is
the given line of symmetry and the blue ring is the hole already punched.

ANSWER

For each figure, measure how far the given hole is from the mirror line, and mark the new hole
at the same distance on the other side, on the line through the hole at right angles to the
mirror.

(a) 1 new hole (b) 1 new hole (c) 1 new hole (d), (e) 1 new hole

Blue = the hole given in the book, yellow = the hole you must mark. Its distance from the dashed
mirror line is the same.

(a) Square with a diagonal mirror — the hole is near the left edge, close to the top-left
corner. Its mirror lies just as far below the top edge as the given one is from the left edge, so
the new hole sits near the top edge. (1 new hole.)
(b) Rectangle with a horizontal mirror — the hole is below the line, near the right edge.
The new hole goes directly above it, the same distance above the line. (1 new hole.)
(c) Triangle with a vertical mirror — the hole is just left of the line, so the new hole is just
right of the line, at the same height. (1 new hole.)
(d) and (e) Circle with a slanting diameter — drop a perpendicular from the hole to the
diameter, continue it the same distance beyond, and punch there. (1 new hole in each.)

Page 23 of 88

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Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
The rule of a mirror: a point and its image are always on a line perpendicular to the

m as e
l
mirror, at equal distances from it. Nothing else is needed to find the second hole.

m .co a g
l a se
a g
Q3 Here are some questions on paper cutting. Consider a vertical fold. We represent it

co m
ag
this way: (Vertical Fold). Similarly, a horizontal fold is represented as follows:

m .
e
(Horizontal Fold).

g l as
ANSWER a
This part only explains the two pictures that will be used in question 4. Learn to read them:
co m
e m.
c o m WHAT IT MEANS g l as
.
PICTURE WHERE THE WHAT HAPPENS TO A CUT

m a
ase
CREASE IS

aglVertical the right half of the sheet is a vertical line every cut is copied to the left and

s
Fold turned over onto the left half through the right — the opened sheet has a

m a
.co agl
(or the left onto the right) middle vertical line of symmetry

a s em
gl (or the
Horizontal the top half is turned down a horizontal line every cut is copied above and
Fold
bottom up)
a
onto the bottom half through the
middle
below — the opened sheet has a
horizontal line of symmetry

co m
m .
e
After one fold the paper is 2 layers thick, so a single cut produces 2 identical marks. After two
m l as
.co g
folds it is 4 layers thick and one cut produces 4 marks.

em a
a s
agl Tip: the crease always becomes a line of symmetry of the opened sheet. Cuts that
m
touch the crease open out into one big hole; cuts on a free edge open out as two

a se
com l
separate notches.
. a g
m
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 24 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q4 After each of the following cuts, predict the shape of the hole when the paper is
opened. After you have made your prediction, make the cutouts and verify your
answer.

a. b.

c.

d.

Question 4, page 225 — the four cuts. The thick edge of each sheet is the fold; the
dashed or dotted line shows where the scissors go. In c and d the last drawing is the
still-folded paper after the cut.

ANSWER

In every case the answer is found the same way: take the piece cut away and add its mirror
image in the crease.

Page 25 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

(a) four-pointed hole (b) ribbon-shaped hole (d) hole + 2 notches

What the sheets look like when they are opened out. The dashed red line is the crease.

(a) The cut piece touches the fold and has a V-shaped dent on its far side. On opening you
get a single hole in the middle of the sheet with two points at the top, two points at the
bottom and a V-dent on each side — a butterfly-like hole. The crease is its line of
symmetry.
(b) A band with a swallow-tail (a “<” notch) is cut from the fold. Opening gives a long ribbon-
shaped hole right across the middle, with a V-notch at each end. This hole has two lines
of symmetry — the crease, and the horizontal line along the middle of the band.
(c) Here the sheet is folded twice (once vertically, then once horizontally), so it is 4 layers
thick. The two small square notches cut on the folded corner open out into four notches,
arranged symmetrically about both creases; the notch that lay on a crease becomes a hole of
double size. The opened sheet has both a vertical and a horizontal line of symmetry.
(d) One vertical fold. The bracket cut on the folded edge opens into a rectangular hole in
the middle of the sheet, and the bracket cut on the free edge opens into two notches, one
on the left edge and one on the right edge.

Check it yourself: do it with a rough sheet of paper before you look at the answer
— predicting first is the whole point of the question.

Page 26 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q5 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. b. The hole in the
centre is a square.

a. b.

Question 5, page 226 — the two sheets you have to end up with. The white part in the
middle of each is the hole.

Note: For the above two questions, check if the 4-sided figures in the centre satisfy
both the properties of a square.

ANSWER

Fold the sheet twice so that the centre of the sheet comes to one corner of the folded packet.
That corner is the closed corner — the centre of the paper. One straight cut near it removes a
piece from all 4 layers at once, and opens out as a hole in the middle.

(a) straight cut (b) slanting cut
Both holes sit at the crossing point of the two creases — the centre of the sheet.

Page 27 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

(a) Upright square hole

1. Fold the sheet in half horizontally, then fold it in half vertically. The centre of the sheet is
now the closed corner of the small square packet.
2. Cut a small square notch at that closed corner, with its two straight sides running along
the two creases.
3. Open out — the four quarter-notches join to make one square hole in the centre.

(b) Square hole standing on a corner (a “diamond”)

1. Fold horizontally and then vertically, exactly as before.
2. Now make a single slanting straight cut across the closed corner, cutting off a small right-
angled triangle with its two equal short sides on the creases.
3. Open out — the four triangles together form a square hole turned through 45°.

Checking the note: in (a) all four sides of the hole are equal and every angle is 90°
— both properties (S1 and S2) hold, so it is a square. In (b), the cut is made at 45° to
both creases and equally far along each, so all four sides are equal and the corners
are right angles again. It is the same square, only tilted; turning a figure never
changes its sides or its angles.

Page 28 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
How many lines of symmetry do these shapes have? a.
e
Q6

m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
.co a g
a s em
a gl Question 6a, page 226 — a square standing on a corner, and an eight-pointed star.

m a s
.co agl
b. A triangle with equal sides and equal angles.

se m
g l a
a

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
m . co agl
l a se
ag page 226 — a triangle with equal sides and equal angles.
Question 6b,

co m
m .
m ase
.co


a g l Page 29 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

c. A hexagon with equal sides and equal angles.

Question 6c, page 227 — a hexagon with equal sides and equal angles.

ANSWER

square: 4 8-pointed star: 8 equilateral triangle: 3 regular hexagon: 6

Every dashed line is a line of symmetry. Notice how a regular figure of n sides (or n points) always has
exactly n of them.

a. The first shape is a square standing on one corner — turning a square does not change it,
so it still has 4 lines of symmetry: two through opposite corners and two through the
midpoints of opposite sides.

Page 30 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

The second shape is a regular eight-pointed star — it has 8 lines of symmetry: 4 through
opposite points and 4 through opposite inner corners.
b. A triangle with equal sides and equal angles is an equilateral triangle. It has 3 lines of
symmetry — one from each corner to the midpoint of the opposite side.
c. A hexagon with equal sides and equal angles is a regular hexagon. It has 6 lines of
symmetry — 3 joining opposite corners and 3 joining the midpoints of opposite sides.

The pattern: a regular polygon with n sides has exactly n lines of symmetry. If n is
odd, each line joins a corner to the midpoint of the opposite side; if n is even, half
the lines join opposite corners and half join midpoints of opposite sides.

Page 31 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q7 Trace each figure and draw the lines of symmetry, if any:

The eight figures printed with this question on page 227 — four made of rhombuses and
four drawn on squared paper.

ANSWER

The four rhombus (diamond) patterns:

Page 32 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

FIGURE LINES OF WHERE
SYMMETRY

Three rhombuses — one 1 the vertical line through the top rhombus
on top, two below

Three rhombuses in a row 2 the vertical line through the middle rhombus and
the horizontal line through all three centres

Four rhombuses making 2 the vertical and the horizontal line through the
one big rhombus centre

Four rhombuses in two 2 the vertical and the horizontal line through the
slanting pairs centre

The four figures drawn on squared paper:

4 lines 2 lines 1 line 4 lines

The four grid figures of page 227 with every line of symmetry marked.

Square with a smaller square inside, divided into four — 4 lines of symmetry (vertical,
horizontal, both diagonals). The inner square sits exactly in the middle, so it does not spoil
any of them.
The eight-sided figure (octagon-like) — 2 lines of symmetry (vertical and horizontal). It is
taller than it is wide, so the diagonals are not lines of symmetry.
The five-sided figure — 1 line of symmetry, the horizontal line through the left corner.
The four-pointed star — 4 lines of symmetry (vertical, horizontal and both diagonals).

Page 33 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
Find the lines of symmetry for the kolam below.
e
Q8

m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
The kolam printed on page 228 — seven six-pointed stars with rhombuses between

m
them.

a se
. com a g l
m
gl ase
a
ANSWER

The kolam is built from seven six-pointed stars — one in the middle and six around it — with
small rhombuses filling the gaps. That makes it a hexagonal design.
co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 34 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

A simplified sketch of the kolam: a star in the centre and six around it, with the 6 lines of symmetry
drawn.

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.

The kolam also turns onto itself at 60°, 120°, 180°, 240°, 300° and 360° — 6 angles of

symmetry.

Page 35 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Did you know? Kolam and rangoli artists use exactly this idea: they draw only one-
sixth (or one-fourth) of the design carefully, and then repeat it around the centre.
That is why these floor drawings look so perfect.

Q9 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?

ANSWER

(a) isosceles — 1 line (b) equilateral — 3 lines (c) scalene — no line
Triangles with exactly one, exactly three and no lines of symmetry.

a. Draw an isosceles triangle — two sides equal, the third different (say 6 cm, 6 cm and 4
cm). Its only line of symmetry runs from the top corner to the midpoint of the unequal side.
b. Draw an equilateral triangle — all three sides equal (say 5 cm each). It has 3 lines of
symmetry.
c. Draw a scalene triangle — all three sides different (say 4 cm, 6 cm and 7 cm). It has no
line of symmetry.

Is a triangle with exactly two lines of symmetry possible? No, it is impossible.

Why: each line of symmetry of a triangle makes two of its sides equal.

One line of symmetry → exactly two sides equal (isosceles).
If there were a second line, another pair of sides would also become equal — and
then all three sides would be equal, which gives an equilateral triangle with three
lines, not two.

So the number of lines of symmetry of a triangle can only be 0, 1 or 3 — never 2.

Page 36 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q10 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.

ANSWER

(a) semicircle — 1 line (b) oval — 2 lines (c) flower — 4 lines
Curved figures with exactly 1, 2 and 4 lines of symmetry.

a. Exactly one line: a semicircle (half a chapati). Its only line of symmetry is the
perpendicular to the straight edge through its midpoint. A “D” shape, a heart or an ice-cream
cone with a rounded top will also do.
b. Exactly two lines: an oval (ellipse), or a rectangle with the two short sides replaced by
semicircular bumps. The lines of symmetry are the long axis and the short axis.
c. Exactly four lines: a flower with four equal petals drawn round a centre, or a square
with a semicircular bump on each side. The lines are the vertical, the horizontal and the two
diagonals.

Careful: a full circle will not do for (c) — it has infinitely many lines of symmetry, not
exactly four.

Page 37 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q11 Copy the following on squared paper. Complete them so that the blue line is a line
of symmetry. Problem (a) has been done for you. Hint: For (c) and (f), see if rotating
the book helps!

(a) (b) (c)

(d) (e) (f)

The six drawings on squared paper, pages 228 and 229. The blue line has to become a
line of symmetry; (a) has been done — the dashed part is the mirror copy.

ANSWER

Work square by square on the grid. For every corner of the given red shape, count the number
of squares from the blue line, go the same number of squares on the other side, and mark
the matching corner. Then join your new corners in the same order.

solid = given

dashed = your mirror copy

blue = line of symmetry

Page 38 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co
How part (a) was completed — every point is copied to the same distance on the other side of the
m
e m.
blue line.

m l as
.co a g
s em
(a) Done in the book: the half-arrow on the right of the vertical blue line is copied to the left,
a
l a kite-shaped figure.
ggiving
a (b) The blue line is horizontal. Copy the stepped shape downwards: every point that is 3

co m
squares above the line goes 3 squares below it. The finished figure looks like the same

m . ag
se
“tower” standing on its own reflection.

l a
ag again. On a diagonal mirror, a point that is a squares right
(c) The blue line is a slanting (diagonal) line. Turn your book so that this line becomes
vertical — then the copying is easy
and b squares up from a point of the line comes back as b squares right and a squares up.

co m
m.
(d) Vertical blue line, small hook on the left — copy it to the right; the two hooks together

m
make a shape like a bracket “]—[”.
o l a se
.c a g — copy it above the
m
(e) Horizontal blue line at the top with a five-sided shape hanging below

l a se to get a closed figure shaped like a diamond with two flat ends.
line
a g (f) Slanting blue line again; rotate the book, then reflect the hexagon-like shape to the other
s
side of the line.
m a
em
.co agl
a s
Why the hint works: a mirror line does not care which way the page is held. Turning

agl of squares easier for your eyes — the finished
the book only makes the counting
answer is the same.

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m. a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 39 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q12 Copy the following drawing on squared paper. Complete each one of them so that
the resulting figure has the two blue lines as lines of symmetry.

(a) (b) (c)

(d) (e) (f)

The six drawings on squared paper, page 229. The two blue lines have to become the
lines of symmetry.

ANSWER

Do the reflection twice:

1. Reflect the given red piece in the first blue line.
2. Now reflect both pieces (the given one and the new one) in the second blue line.

You will end up with 4 copies of the piece, one in each of the four regions made by the two
lines.

Page 40 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

One given piece (solid) becomes four (dashed copies) when both blue lines must be lines of
symmetry.

(a) The two blue lines are the diagonals. The short red segment near the bottom must be
copied into all four quarters; the finished figure looks like a square standing on a corner with
four equal marks.
(b) Diagonals again, with a zigzag on the left. Reflect it in one diagonal, then reflect the pair
in the other — a four-armed “windmill with mirrors”.
(c), (d), (e), (f) Here the blue lines are the vertical and the horizontal line. Reflect the given
piece across the vertical line, then reflect both across the horizontal line, giving 4 identical
pieces placed symmetrically about the centre.

Page 41 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

A bonus fact: whenever a figure has two lines of symmetry crossing at right angles,
it automatically has rotational symmetry of order 2 — a half turn (180°) about the
crossing point brings it back onto itself.

Q13 Copy the following on a dot grid. For each figure draw two more lines to make a
shape that has a line of symmetry.

(a) (b) (c)

(d) (e) (f)

The six line figures drawn on a dot grid, page 230.

ANSWER

Many different answers are possible. The easiest method is:

1. Choose the line that you want as the axis — usually a vertical, a horizontal or a 45° line
through the dots.
2. Reflect the given lines in it. Because two of the given corners usually fall on the axis, exactly
two new lines are needed to close the shape.

Page 42 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

One of the six figures completed: the two red dashed lines are the two new lines, and the green line is
the line of symmetry.

Here is one working answer for each of the six figures (using columns numbered 0–5 from the
left and rows 0–5 from the top):

Page 43 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
FIGURE LINES ALREADY TWO LINES LINE OF SYMMETRY

as e
com
DRAWN TO ADD

.(4,0)–(0,1)–(4,5) a g l
e m
as
1 (4,5)–(5,1) and the slanting line through the midpoints of the

agl
(5,1)–(4,0) two parallel sides (an isosceles trapezium)

2 (2,0)–(0,3)–(1,5)– (3,5)–(4,3) and the vertical line through (2,0)

co m
. ag
(3,5) (4,3)–(2,0)

se m
l a
ag
3 (0,0)–(2,1)–(4,0)– (2,5)–(0,2) and the vertical line through (2,0)
(4,2)–(2,5) (0,2)–(0,0)

4 (0,0)–(4,0)–(4,1)– (1,4)–(0,4) and the 45° diagonal starting at (0,0)
co m
se m.
(1,4) (0,4)–(0,0)

m l a
5
m .co (2,3)–(0,1)–(2,5) ag row 3
the horizontal line through

se
(0,5)–(2,3) and

l a
(0,5)–(2,1)

a 6g (0,0)–(2,0)–(5,2)– (2,4)–(0,4) and the horizontal line through row 2

m a s
agl
(2,4) (0,4)–(0,0)

m .co
l a se
a g
Check it yourself: after drawing, hold a small mirror upright along the line you
chose. If the half you can see plus its reflection looks exactly like your whole figure,
the answer is right.
co m
m .
m as e
.co a g l
m
ase Questions — Pages 230 & 231
agl
In-text

se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 44 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Section 9.2 Rotational Symmetry

Q1 Will the windmill above look exactly the same when rotated through an angle of
less than 90°?

The paper windmill on page 230. The red point at the centre is the centre of rotation.

ANSWER

No. The paper windmill has four identical blades placed at equal distances round the centre.

Angle between two neighbouring blades = 360° ÷ 4 = 90°

If you turn it by anything less than 90° — say 30° or 45° or 80° — every blade stops between two
old blade positions, so the picture looks tilted and does not match the original.

Page 45 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Only when the turn reaches 90° does each blade land exactly where the next blade was, and the
windmill looks untouched.

Why 90° and not less: for the figure to look the same, a blade must land on a blade.
The first chance to do that comes after a turn of one whole “gap”, and the gap is 90°.
That is why 90° is called the smallest angle of symmetry of the windmill.

Page 46 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q2 Thus, we see that the windmill has 4 angles of symmetry. Do you know of any other
shape that has exactly four angles of symmetry?

The paper windmill on page 230. The red point at the centre is the centre of rotation.

ANSWER

Yes, many. Any figure built by repeating one piece four times around a centre has exactly 4
angles of symmetry — 90°, 180°, 270°, 360°.

A square (and the square tile of a chessboard).
A plus sign “+” made of four equal arms, like the figure on page 232.
A four-petal flower and the four-pointed star of page 227.
A table fan with 4 blades, a ceiling fan with 4 blades, a car steering wheel with 4 equal
spokes.

Page 47 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Many rangoli designs, and the shape of a playing-card “club” arrangement of 4 identical
marks.

Careful: the square has 4 angles of symmetry and 4 lines of symmetry, but the
windmill has 4 angles of symmetry and no line of symmetry. Angles and lines are
counted separately.

Q3 How many angles of symmetry does a square have? How much rotation does it
require to get the initial square? Do you know where to mark the centre of
rotation? What are the other angles of symmetry?

ANSWER

A square has 4 angles of symmetry.

A B D A

90°
→

D C C B

A quarter turn (90°) clockwise about the red centre: A goes to B's place, B to C's, C to D's and D to A's.
The square looks unchanged.

How much rotation is needed? A turn of 90° is enough — the square already covers itself.
Each corner simply moves to the next corner's place: A → B, B → C, C → D, D → A.
Where is the centre of rotation? At the centre of the square — the point where its two
diagonals cross. It is the only point that does not move.
The other angles of symmetry:

90° (quarter turn), 180° (half turn), 270° (three-quarter turn), 360° (full turn)

Page 48 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
Why exactly these four: after 180° the corners read C, D, A, B; after 270° they read

m l a se
D, A, B, C; after 360° everything is back where it started. Any other angle would leave
o g
.c between two positions, and the square would look tilted.
a
m
the corners

l a se
ag
In-text Questions — Pages 232 to 235 om
e m .c ag
a s
agl

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 49 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Rotational Symmetry of Figures with Radial Arms

TRY THIS

Q1 Consider this figure, a picture with 4 radial arms. How many angles of symmetry
does it have? What are they? Note that the angle between adjacent central dotted
lines is 90°.

90°

The figure with 4 radial arms, page 232. The angle between two adjacent dotted lines is
90°.

ANSWER

The four arms are identical and the angle between neighbouring arms is 90°, so each arm can
take the place of the next one.

Page 50 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

90°

Four radial arms, 90° apart. Every quarter turn brings the figure back onto itself.

Angles of symmetry = 90°, 180°, 270°, 360° → 4 angles

Why: a turn of 90° carries arm 1 onto arm 2, arm 2 onto arm 3, arm 3 onto arm 4
and arm 4 onto arm 1 — the picture is unchanged. The same is true of 180°, 270°
and the full turn of 360°.

Page 51 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q2 Can you change the angles between the radial arms so that the figure still has 4
angles of symmetry? Try drawing it.

90°

The figure with 4 radial arms, page 232. The angle between two adjacent dotted lines is
90°.

ANSWER

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.

Smallest angle of symmetry = 360° ÷ 4 = 90°

Page 52 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Why: if the figure looks the same after a turn of 90°, that turn must carry each arm
onto the next arm. So the gaps between consecutive arms must all be equal, and
four equal gaps in a full turn means 90° each. If you make one gap 80° and another
100°, the arms no longer match after a quarter turn and only the 360° turn is left.

What you can change: the arms themselves. Make them longer, shorter, thicker, curved like a
fan blade or shaped like a leaf — as long as all four are identical and 90° apart, the figure still
has exactly 4 angles of symmetry. Try drawing a four-blade fan and a four-petal flower and
check them with a paper cut-out.

Page 53 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
How will you modify the figure above so that it has only two angles of symmetry?
e
Q3

m l as
m .co a g
l a se
ag

co m
e m . ag
g l as
a
90°
co m
em.
m l as
m .co a g
l a se
ag
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
o m l a
m .c 232. The angle between two adjacent dotted lines is ag
The figure with 4 radial arms, page

l a se 90°.

ag

co m
ANSWER
m .
as e
com arms different. g l
Make the arms unequal in pairs: keep the two opposite arms alike, and make the other two
. a
sem
opposite
a
agl c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 54 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Two long arms and two short arms. A quarter turn no longer matches — only a half turn does.

Angles of symmetry = 180°, 360° → only 2 angles

A turn of 90° would try to put a long arm on a short arm — it fails.
A turn of 180° puts each arm on its opposite arm, which is identical — it works.

Another easy way: keep the arms equal but add a small bent flag at the end of each
arm, all bending the same way round (the shape shown in the book). A half turn still
matches, a quarter turn does not.

Page 55 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Q4 Let us try with 3 radial arms as in the figure below. How many angles of symmetry
does it have and what are they? Trace and cut out a copy of this figure. By rotating
the cutout over this figure determine its angles of rotation.

The figure with 3 radial arms on page 233 — the three arms are not equally spaced.

ANSWER

Cut out a tracing and turn it slowly over the printed figure. You will find that no partial turn
ever matches.

Angle of symmetry = 360° only → 1 angle

So this figure does not have rotational symmetry, even though it has three arms.

Page 56 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

Why it fails: in the printed figure the three arms are not equally spaced — the
angles between them are all different. For a turn to bring arm 1 onto arm 2, the gap
between arm 1 and arm 2 must be the same as the gap between arm 2 and arm 3
and between arm 3 and arm 1. That is not the case here, so only the full turn of 360°
works.

Remember: 360° is an angle of symmetry of every figure. A figure is said to have
rotational symmetry only when it has some angle of symmetry smaller than 360°.

Q5 However, can anything in the figure be changed to make it have 3 angles of
symmetry? Can it be done by changing the angles between the dotted lines?
Observe that ∠A must overlap ∠B, ∠B must overlap ∠C and ∠C must overlap ∠A.
So, ∠A = ∠B = ∠C. What must this angle be? Now how many angles of rotation does
the figure have and what are they?

C A B C

B A

The two rough diagrams on page 234 — the figure and a rotated copy of it.

ANSWER

Yes — change the angles between the arms so that all three are equal.

∠A = ∠B = ∠C

∠A + ∠B + ∠C = 360° (one full turn)

3 × ∠A = 360°

∠A = 360° ÷ 3 = 120°

Page 57 of 88

Class 6 Maths Chapter 9 Symmetry AglaSem · NCERT Solutions

120°

Three equal arms with 120° between neighbouring arms — now the figure has rotational symmetry.

With 120° between the adjacent dotted lines, the figure has 3 angles of symmetry:

120° (one third of a turn), 240° (120° + 120°), 360° (120° + 120° + 120°)

Why exactly these: a turn of 120° sends arm A to arm B, B to C and C to A. Doing it
twice (240°) sends A to C, and doing it three times (360°) brings everything home.
Notice that all the angles of symmetry are multiples of the smallest one, 120°.

Page 58 of 88

as e
Class 6 Maths Chapter 9 Symmetry
a g l AglaSem · NCERT Solutions

co m
m.
Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b)
se
Q6

o m l a
6 angles of symmetry? Also find the angles of symmetry in each case. Hint: Use 5
radial.c g adjacent radial
arms for the first case. What should the angle betweenatwo
m
se be?
g l aarms
a
m
ANSWER

. co ag
m
Use as many equal arms as the number of angles you want, and share the full turn of 360°
equally between them.
l a se
agdraw 5 equal arms.
a) Exactly 5 angles of symmetry —

co m
Angle between two adjacent arms = 360° ÷ 5 = 72°
em.
m l as
.co
Angles of symmetry = 72°, 144°, 216°, 288°, 360°
m a g
l a se
g
a b) Exactly 6 angles of symmetry — draw 6 equal arms.
m a s
m .co agl
se
Angle between two adjacent arms = 360° ÷ 6 = 60°

g l a
a
Angles of symmetry = 60°, 120°, 180°, 240°, 300°, 360°

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m 5 arms, 72° ase 60°
6 arms,
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l a se Equal arms spread equally round the centre: 5 arms give 72°, 6 arms give 60°.

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The general rule: a figure with n equal arms, equally spaced, has n angles of

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symmetry, and its smallest angle of symmetry is 360° ÷ n. All the other angles are
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