NCERT Textbook: Symmetry | Mathematics (Maths) Class 6 PDF Download

 Page 1


SYMMETRY
9
Look around you — you may find many objects that catch your 
attention. Some such things are shown below:
Butterfly Flower  
Pinwheel  Rangoli  
There is something beautiful about the pictures above. 
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?
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? In the 
Chapter 9_Symmetry.indd   217 10-08-2024   16:20:09
Page 2


SYMMETRY
9
Look around you — you may find many objects that catch your 
attention. Some such things are shown below:
Butterfly Flower  
Pinwheel  Rangoli  
There is something beautiful about the pictures above. 
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?
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? In the 
Chapter 9_Symmetry.indd   217 10-08-2024   16:20:09
Ganita Prakash | Grade 6
218
rangoli, the red petals come back onto themselves when the flower 
is rotated by 90° around the centre and so do the other parts of the 
rangoli.
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? What about 
the other pictures — what is it about 
those structures that appeals to you and 
what are the patterns in those structures 
that repeat?
On the other hand, look at this picture 
of clouds. There is no such repetitive pattern.
We can say that the first four figures are symmetrical. and the last one is 
not symmetrical. A symmetry refers to a part or parts of a figure that 
are repeated in some definite pattern.
Taj Mahal Gopuram
What are the symmetries that you see in these beautiful structures?
Clouds
Chapter 9_Symmetry.indd   218 10-08-2024   16:20:10
Page 3


SYMMETRY
9
Look around you — you may find many objects that catch your 
attention. Some such things are shown below:
Butterfly Flower  
Pinwheel  Rangoli  
There is something beautiful about the pictures above. 
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?
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? In the 
Chapter 9_Symmetry.indd   217 10-08-2024   16:20:09
Ganita Prakash | Grade 6
218
rangoli, the red petals come back onto themselves when the flower 
is rotated by 90° around the centre and so do the other parts of the 
rangoli.
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? What about 
the other pictures — what is it about 
those structures that appeals to you and 
what are the patterns in those structures 
that repeat?
On the other hand, look at this picture 
of clouds. There is no such repetitive pattern.
We can say that the first four figures are symmetrical. and the last one is 
not symmetrical. A symmetry refers to a part or parts of a figure that 
are repeated in some definite pattern.
Taj Mahal Gopuram
What are the symmetries that you see in these beautiful structures?
Clouds
Chapter 9_Symmetry.indd   218 10-08-2024   16:20:10
Symmetry
219
9.1 Line of Symmetry
Figure (a) shows the picture of a blue triangle with a dotted line. 
What if you fold the triangle along the dotted line? Yes, one half 
of the triangle covers the other half completely. These are called 
mirror halves! 
(a)
(b)
What about Figure (b) with the four puzzle pieces and a dotted line 
passing through the middle? Are they mirror halves? No, when we fold 
along the line, the left half does not exactly fit over the righ t half.
A line that cuts a figure into two  parts that exactly overlap when 
folded along that line is called a line of symmetry of the figure.
 Figure it Out 
1. Do you see any line of symmetry in the figures at the start of the 
chapter? What about in the picture of the cloud?
2. For each of the following figures, identify the line(s) of symmetry 
if it exists.
Chapter 9_Symmetry.indd   219 10-08-2024   16:20:11
Page 4


SYMMETRY
9
Look around you — you may find many objects that catch your 
attention. Some such things are shown below:
Butterfly Flower  
Pinwheel  Rangoli  
There is something beautiful about the pictures above. 
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?
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? In the 
Chapter 9_Symmetry.indd   217 10-08-2024   16:20:09
Ganita Prakash | Grade 6
218
rangoli, the red petals come back onto themselves when the flower 
is rotated by 90° around the centre and so do the other parts of the 
rangoli.
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? What about 
the other pictures — what is it about 
those structures that appeals to you and 
what are the patterns in those structures 
that repeat?
On the other hand, look at this picture 
of clouds. There is no such repetitive pattern.
We can say that the first four figures are symmetrical. and the last one is 
not symmetrical. A symmetry refers to a part or parts of a figure that 
are repeated in some definite pattern.
Taj Mahal Gopuram
What are the symmetries that you see in these beautiful structures?
Clouds
Chapter 9_Symmetry.indd   218 10-08-2024   16:20:10
Symmetry
219
9.1 Line of Symmetry
Figure (a) shows the picture of a blue triangle with a dotted line. 
What if you fold the triangle along the dotted line? Yes, one half 
of the triangle covers the other half completely. These are called 
mirror halves! 
(a)
(b)
What about Figure (b) with the four puzzle pieces and a dotted line 
passing through the middle? Are they mirror halves? No, when we fold 
along the line, the left half does not exactly fit over the righ t half.
A line that cuts a figure into two  parts that exactly overlap when 
folded along that line is called a line of symmetry of the figure.
 Figure it Out 
1. Do you see any line of symmetry in the figures at the start of the 
chapter? What about in the picture of the cloud?
2. For each of the following figures, identify the line(s) of symmetry 
if it exists.
Chapter 9_Symmetry.indd   219 10-08-2024   16:20:11
Ganita Prakash | Grade 6
220
Figures with more than one line of symmetry
Does a square have only one line of symmetry?
Take a square piece of paper. By folding, find all its lines of symmetry.  
Fold 1
Fold 2
Fold 4
Fold 3
Here are the different folds giving different lines of symmetry.
• Fold the paper into half vertically.
• Fold it again into half horizontally. (i.e. you have folded it 
twice). Now open out the folds.
Vertical Fold
Horizontal Fold
Again fold the square into half (for a third time now), but this 
time along a diagonal, as shown in the figure. Again, open it.
Chapter 9_Symmetry.indd   220 10-08-2024   16:20:13
Page 5


SYMMETRY
9
Look around you — you may find many objects that catch your 
attention. Some such things are shown below:
Butterfly Flower  
Pinwheel  Rangoli  
There is something beautiful about the pictures above. 
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?
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? In the 
Chapter 9_Symmetry.indd   217 10-08-2024   16:20:09
Ganita Prakash | Grade 6
218
rangoli, the red petals come back onto themselves when the flower 
is rotated by 90° around the centre and so do the other parts of the 
rangoli.
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? What about 
the other pictures — what is it about 
those structures that appeals to you and 
what are the patterns in those structures 
that repeat?
On the other hand, look at this picture 
of clouds. There is no such repetitive pattern.
We can say that the first four figures are symmetrical. and the last one is 
not symmetrical. A symmetry refers to a part or parts of a figure that 
are repeated in some definite pattern.
Taj Mahal Gopuram
What are the symmetries that you see in these beautiful structures?
Clouds
Chapter 9_Symmetry.indd   218 10-08-2024   16:20:10
Symmetry
219
9.1 Line of Symmetry
Figure (a) shows the picture of a blue triangle with a dotted line. 
What if you fold the triangle along the dotted line? Yes, one half 
of the triangle covers the other half completely. These are called 
mirror halves! 
(a)
(b)
What about Figure (b) with the four puzzle pieces and a dotted line 
passing through the middle? Are they mirror halves? No, when we fold 
along the line, the left half does not exactly fit over the righ t half.
A line that cuts a figure into two  parts that exactly overlap when 
folded along that line is called a line of symmetry of the figure.
 Figure it Out 
1. Do you see any line of symmetry in the figures at the start of the 
chapter? What about in the picture of the cloud?
2. For each of the following figures, identify the line(s) of symmetry 
if it exists.
Chapter 9_Symmetry.indd   219 10-08-2024   16:20:11
Ganita Prakash | Grade 6
220
Figures with more than one line of symmetry
Does a square have only one line of symmetry?
Take a square piece of paper. By folding, find all its lines of symmetry.  
Fold 1
Fold 2
Fold 4
Fold 3
Here are the different folds giving different lines of symmetry.
• Fold the paper into half vertically.
• Fold it again into half horizontally. (i.e. you have folded it 
twice). Now open out the folds.
Vertical Fold
Horizontal Fold
Again fold the square into half (for a third time now), but this 
time along a diagonal, as shown in the figure. Again, open it.
Chapter 9_Symmetry.indd   220 10-08-2024   16:20:13
Symmetry
221
Fold it into half (for the fourth time), but this time along the other 
diagonal, as shown in the figure. Open out the fold.
 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?
 Thus, figures can have multiple lines of symmetry. The figures 
below also have multiple lines of symmetry. Can you find them all?.
 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?
Reflection
So far we have been saying that when we fold a figure along a line of 
symmetry, the two parts overlap completely. We could also say that 
the part of the figure on one side of the line of symmetry is reflected 
by the line to the other side; similarly, the part of the figure on the 
other side of the line of symmetry is reflected to the first side! Let us 
understand this by labeling some points on the figure.
The figure shows a square with its corners labeled A, B, C and D. 
Let us first consider the vertical line of symmetry. When we reflect 
Chapter 9_Symmetry.indd   221 10-08-2024   16:20:13