NCERT Textbook: Understanding Elementary Shapes | Mathematics (Maths) Class 6 PDF Download

 Page 1


All the shapes we see around us are formed using curves or lines. We can see
corners, edges, planes, open curves and closed curves in our surroundings.
We organise them into line segments, angles, triangles, polygons and circles.
We find that they have different sizes and measures. Let us now try to develop
tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three,
a quadrilateral of four line segments.
A line segment is a fixed portion of a line. This makes it possible to measure
a line segment. This measure of each line segment is a unique number called
its “length”. We use this idea to compare line segments.
To compare any two line segments, we find a relation between their lengths.
This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you
tell which one is longer?
You can see that AB is
longer.
But you cannot always be
sure about your usual judgment.
For example, look at the
adjoining segments :
5.1 Introduction
Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5
Understanding Understanding
Understanding Understanding Understanding
Elementary Elementary
Elementary Elementary Elementary
Shapes Shapes
Shapes Shapes Shapes
Rationalised 2023-24
Page 2


All the shapes we see around us are formed using curves or lines. We can see
corners, edges, planes, open curves and closed curves in our surroundings.
We organise them into line segments, angles, triangles, polygons and circles.
We find that they have different sizes and measures. Let us now try to develop
tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three,
a quadrilateral of four line segments.
A line segment is a fixed portion of a line. This makes it possible to measure
a line segment. This measure of each line segment is a unique number called
its “length”. We use this idea to compare line segments.
To compare any two line segments, we find a relation between their lengths.
This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you
tell which one is longer?
You can see that AB is
longer.
But you cannot always be
sure about your usual judgment.
For example, look at the
adjoining segments :
5.1 Introduction
Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5
Understanding Understanding
Understanding Understanding Understanding
Elementary Elementary
Elementary Elementary Elementary
Shapes Shapes
Shapes Shapes Shapes
Rationalised 2023-24
MATHEMATICS
60
The difference in lengths between these two may not be obvious. This makes
other ways of comparing necessary.
In this adjacent figure, AB and PQ have the  same
lengths. This is not quite obvious.
So, we need better methods of comparing line
segments.
(ii) Comparison by Tracing
To compare AB and CD , we use a tracing paper, trace CD and place the
traced segment on AB .
Can you decide now which one among AB and CD is longer?
The method depends upon the accuracy in tracing the line segment.
Moreover, if you want to compare with another length, you have to trace
another line segment. This is difficult and  you cannot trace the lengths
everytime you want to compare them.
(iii) Comparison using Ruler and a Divider
Have you seen or can you recognise all the instruments  in your
instrument box? Among other things, you have a ruler and a divider.
Note how the ruler is marked along one of its edges.
It is divided into 15 parts. Each of these 15 parts is of
length 1cm.
Each centimetre is divided into 10subparts.
Each subpart of the division of a cm is 1mm.
How many millimetres make
one centimetre? Since 1cm = 10
mm, how will we write 2 cm?
3mm? What do we mean
by 7.7 cm?
Place the zero mark of the ruler at A. Read the mark against B. This gives the
length of AB . Suppose the length is 5.8 cm, we may write,
Length AB = 5.8 cm or more simply as AB = 5.8 cm.
There is room for errors even in this procedure. The thickness of the ruler
may cause difficulties in reading off the marks on it.
Ruler
Divider
1 mm is 0.1 cm.
2 mm is 0.2 cm and so on .
2.3 cm will mean 2 cm
and 3 mm.
Rationalised 2023-24
Page 3


All the shapes we see around us are formed using curves or lines. We can see
corners, edges, planes, open curves and closed curves in our surroundings.
We organise them into line segments, angles, triangles, polygons and circles.
We find that they have different sizes and measures. Let us now try to develop
tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three,
a quadrilateral of four line segments.
A line segment is a fixed portion of a line. This makes it possible to measure
a line segment. This measure of each line segment is a unique number called
its “length”. We use this idea to compare line segments.
To compare any two line segments, we find a relation between their lengths.
This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you
tell which one is longer?
You can see that AB is
longer.
But you cannot always be
sure about your usual judgment.
For example, look at the
adjoining segments :
5.1 Introduction
Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5
Understanding Understanding
Understanding Understanding Understanding
Elementary Elementary
Elementary Elementary Elementary
Shapes Shapes
Shapes Shapes Shapes
Rationalised 2023-24
MATHEMATICS
60
The difference in lengths between these two may not be obvious. This makes
other ways of comparing necessary.
In this adjacent figure, AB and PQ have the  same
lengths. This is not quite obvious.
So, we need better methods of comparing line
segments.
(ii) Comparison by Tracing
To compare AB and CD , we use a tracing paper, trace CD and place the
traced segment on AB .
Can you decide now which one among AB and CD is longer?
The method depends upon the accuracy in tracing the line segment.
Moreover, if you want to compare with another length, you have to trace
another line segment. This is difficult and  you cannot trace the lengths
everytime you want to compare them.
(iii) Comparison using Ruler and a Divider
Have you seen or can you recognise all the instruments  in your
instrument box? Among other things, you have a ruler and a divider.
Note how the ruler is marked along one of its edges.
It is divided into 15 parts. Each of these 15 parts is of
length 1cm.
Each centimetre is divided into 10subparts.
Each subpart of the division of a cm is 1mm.
How many millimetres make
one centimetre? Since 1cm = 10
mm, how will we write 2 cm?
3mm? What do we mean
by 7.7 cm?
Place the zero mark of the ruler at A. Read the mark against B. This gives the
length of AB . Suppose the length is 5.8 cm, we may write,
Length AB = 5.8 cm or more simply as AB = 5.8 cm.
There is room for errors even in this procedure. The thickness of the ruler
may cause difficulties in reading off the marks on it.
Ruler
Divider
1 mm is 0.1 cm.
2 mm is 0.2 cm and so on .
2.3 cm will mean 2 cm
and 3 mm.
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
UNDERSTANDING ELEMENTARY SHAPES
61
Think, discuss and write
1. What other errors and difficulties might we face?
2. What kind of errors can occur if viewing the mark on the ruler is not
proper? How can one avoid it?
Positioning error
To get correct measure, the eye should be
correctly positioned,  just vertically above
the mark. Otherwise errors can happen due
to angular viewing.
Can we avoid this problem? Is there a better way?
Let us use the divider to measure length.
Open the divider. Place the end point of one
of its arms at A and the end point of the second
arm at B. T aking care that opening of the divider
is not disturbed, lift the divider and place it on
the ruler. Ensure that one end point is at the zero
mark of the ruler. Now read the mark against
the other end point.
EXERCISE 5.1
1. What is the disadvantage in  comparing line
segments by mere observation?
2. Why is it better to  use a divider than a ruler, while measuring the length of a line
segment?
3. Draw any line segment, say 
AB
. Take any point C lying in between A and B.
Measure the lengths of AB, BC and AC. Is AB = AC + CB?
[Note : If  A,B,C are any three points on a line such that AC + CB = AB, then we can
be sure that C lies between A and B.]
1. T ake any post card. Use
the above technique to
measure its two
adjacent sides.
2. Select any three objects
having a flat top.
Measure all sides of the
top using a divider and
a ruler.
Rationalised 2023-24
Page 4


All the shapes we see around us are formed using curves or lines. We can see
corners, edges, planes, open curves and closed curves in our surroundings.
We organise them into line segments, angles, triangles, polygons and circles.
We find that they have different sizes and measures. Let us now try to develop
tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three,
a quadrilateral of four line segments.
A line segment is a fixed portion of a line. This makes it possible to measure
a line segment. This measure of each line segment is a unique number called
its “length”. We use this idea to compare line segments.
To compare any two line segments, we find a relation between their lengths.
This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you
tell which one is longer?
You can see that AB is
longer.
But you cannot always be
sure about your usual judgment.
For example, look at the
adjoining segments :
5.1 Introduction
Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5
Understanding Understanding
Understanding Understanding Understanding
Elementary Elementary
Elementary Elementary Elementary
Shapes Shapes
Shapes Shapes Shapes
Rationalised 2023-24
MATHEMATICS
60
The difference in lengths between these two may not be obvious. This makes
other ways of comparing necessary.
In this adjacent figure, AB and PQ have the  same
lengths. This is not quite obvious.
So, we need better methods of comparing line
segments.
(ii) Comparison by Tracing
To compare AB and CD , we use a tracing paper, trace CD and place the
traced segment on AB .
Can you decide now which one among AB and CD is longer?
The method depends upon the accuracy in tracing the line segment.
Moreover, if you want to compare with another length, you have to trace
another line segment. This is difficult and  you cannot trace the lengths
everytime you want to compare them.
(iii) Comparison using Ruler and a Divider
Have you seen or can you recognise all the instruments  in your
instrument box? Among other things, you have a ruler and a divider.
Note how the ruler is marked along one of its edges.
It is divided into 15 parts. Each of these 15 parts is of
length 1cm.
Each centimetre is divided into 10subparts.
Each subpart of the division of a cm is 1mm.
How many millimetres make
one centimetre? Since 1cm = 10
mm, how will we write 2 cm?
3mm? What do we mean
by 7.7 cm?
Place the zero mark of the ruler at A. Read the mark against B. This gives the
length of AB . Suppose the length is 5.8 cm, we may write,
Length AB = 5.8 cm or more simply as AB = 5.8 cm.
There is room for errors even in this procedure. The thickness of the ruler
may cause difficulties in reading off the marks on it.
Ruler
Divider
1 mm is 0.1 cm.
2 mm is 0.2 cm and so on .
2.3 cm will mean 2 cm
and 3 mm.
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
UNDERSTANDING ELEMENTARY SHAPES
61
Think, discuss and write
1. What other errors and difficulties might we face?
2. What kind of errors can occur if viewing the mark on the ruler is not
proper? How can one avoid it?
Positioning error
To get correct measure, the eye should be
correctly positioned,  just vertically above
the mark. Otherwise errors can happen due
to angular viewing.
Can we avoid this problem? Is there a better way?
Let us use the divider to measure length.
Open the divider. Place the end point of one
of its arms at A and the end point of the second
arm at B. T aking care that opening of the divider
is not disturbed, lift the divider and place it on
the ruler. Ensure that one end point is at the zero
mark of the ruler. Now read the mark against
the other end point.
EXERCISE 5.1
1. What is the disadvantage in  comparing line
segments by mere observation?
2. Why is it better to  use a divider than a ruler, while measuring the length of a line
segment?
3. Draw any line segment, say 
AB
. Take any point C lying in between A and B.
Measure the lengths of AB, BC and AC. Is AB = AC + CB?
[Note : If  A,B,C are any three points on a line such that AC + CB = AB, then we can
be sure that C lies between A and B.]
1. T ake any post card. Use
the above technique to
measure its two
adjacent sides.
2. Select any three objects
having a flat top.
Measure all sides of the
top using a divider and
a ruler.
Rationalised 2023-24
MATHEMATICS
62
4. If  A,B,C are three points on a line such that AB = 5 cm, BC = 3 cm and
AC = 8 cm, which one of them lies between the other two?
5. Verify, whether D is the mid point of AG .
6. If B is the mid point of AC and C is the mid
point of BD , where A,B,C,D lie on a straight line, say why AB = CD?
7. Draw five triangles and measure their sides. Check in each case, if the sum of
the lengths of any two sides is always less than the third side.
5.3 Angles – ‘Right’ and ‘Straight’
You have heard of directions in Geography. We know that China is to the
north of India, Sri Lanka is to the south. We also know that Sun rises in the
east and sets in the west. There are four main directions. They are North (N),
South (S), East (E) and West (W).
Do you know which direction is opposite to north?
Which direction is opposite to west?
Just recollect what you know already. We now use this knowledge to learn
a few properties about angles.
Stand facing north.
Turn clockwise to east.
We say, you have turned through a right angle.
Follow this by a ‘right-angle-turn’, clockwise.
You now face south.
If you turn by a right angle in the anti-clockwise
direction, which direction will you face? It is east
again! (Why?)
Study the following positions :
Do This
Y ou stand facing
north
By a ‘right-angle-turn’
clockwise, you now
face east
By another
‘right-angle-turn’ you
finally face south.
Rationalised 2023-24
Page 5


All the shapes we see around us are formed using curves or lines. We can see
corners, edges, planes, open curves and closed curves in our surroundings.
We organise them into line segments, angles, triangles, polygons and circles.
We find that they have different sizes and measures. Let us now try to develop
tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three,
a quadrilateral of four line segments.
A line segment is a fixed portion of a line. This makes it possible to measure
a line segment. This measure of each line segment is a unique number called
its “length”. We use this idea to compare line segments.
To compare any two line segments, we find a relation between their lengths.
This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you
tell which one is longer?
You can see that AB is
longer.
But you cannot always be
sure about your usual judgment.
For example, look at the
adjoining segments :
5.1 Introduction
Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5
Understanding Understanding
Understanding Understanding Understanding
Elementary Elementary
Elementary Elementary Elementary
Shapes Shapes
Shapes Shapes Shapes
Rationalised 2023-24
MATHEMATICS
60
The difference in lengths between these two may not be obvious. This makes
other ways of comparing necessary.
In this adjacent figure, AB and PQ have the  same
lengths. This is not quite obvious.
So, we need better methods of comparing line
segments.
(ii) Comparison by Tracing
To compare AB and CD , we use a tracing paper, trace CD and place the
traced segment on AB .
Can you decide now which one among AB and CD is longer?
The method depends upon the accuracy in tracing the line segment.
Moreover, if you want to compare with another length, you have to trace
another line segment. This is difficult and  you cannot trace the lengths
everytime you want to compare them.
(iii) Comparison using Ruler and a Divider
Have you seen or can you recognise all the instruments  in your
instrument box? Among other things, you have a ruler and a divider.
Note how the ruler is marked along one of its edges.
It is divided into 15 parts. Each of these 15 parts is of
length 1cm.
Each centimetre is divided into 10subparts.
Each subpart of the division of a cm is 1mm.
How many millimetres make
one centimetre? Since 1cm = 10
mm, how will we write 2 cm?
3mm? What do we mean
by 7.7 cm?
Place the zero mark of the ruler at A. Read the mark against B. This gives the
length of AB . Suppose the length is 5.8 cm, we may write,
Length AB = 5.8 cm or more simply as AB = 5.8 cm.
There is room for errors even in this procedure. The thickness of the ruler
may cause difficulties in reading off the marks on it.
Ruler
Divider
1 mm is 0.1 cm.
2 mm is 0.2 cm and so on .
2.3 cm will mean 2 cm
and 3 mm.
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
UNDERSTANDING ELEMENTARY SHAPES
61
Think, discuss and write
1. What other errors and difficulties might we face?
2. What kind of errors can occur if viewing the mark on the ruler is not
proper? How can one avoid it?
Positioning error
To get correct measure, the eye should be
correctly positioned,  just vertically above
the mark. Otherwise errors can happen due
to angular viewing.
Can we avoid this problem? Is there a better way?
Let us use the divider to measure length.
Open the divider. Place the end point of one
of its arms at A and the end point of the second
arm at B. T aking care that opening of the divider
is not disturbed, lift the divider and place it on
the ruler. Ensure that one end point is at the zero
mark of the ruler. Now read the mark against
the other end point.
EXERCISE 5.1
1. What is the disadvantage in  comparing line
segments by mere observation?
2. Why is it better to  use a divider than a ruler, while measuring the length of a line
segment?
3. Draw any line segment, say 
AB
. Take any point C lying in between A and B.
Measure the lengths of AB, BC and AC. Is AB = AC + CB?
[Note : If  A,B,C are any three points on a line such that AC + CB = AB, then we can
be sure that C lies between A and B.]
1. T ake any post card. Use
the above technique to
measure its two
adjacent sides.
2. Select any three objects
having a flat top.
Measure all sides of the
top using a divider and
a ruler.
Rationalised 2023-24
MATHEMATICS
62
4. If  A,B,C are three points on a line such that AB = 5 cm, BC = 3 cm and
AC = 8 cm, which one of them lies between the other two?
5. Verify, whether D is the mid point of AG .
6. If B is the mid point of AC and C is the mid
point of BD , where A,B,C,D lie on a straight line, say why AB = CD?
7. Draw five triangles and measure their sides. Check in each case, if the sum of
the lengths of any two sides is always less than the third side.
5.3 Angles – ‘Right’ and ‘Straight’
You have heard of directions in Geography. We know that China is to the
north of India, Sri Lanka is to the south. We also know that Sun rises in the
east and sets in the west. There are four main directions. They are North (N),
South (S), East (E) and West (W).
Do you know which direction is opposite to north?
Which direction is opposite to west?
Just recollect what you know already. We now use this knowledge to learn
a few properties about angles.
Stand facing north.
Turn clockwise to east.
We say, you have turned through a right angle.
Follow this by a ‘right-angle-turn’, clockwise.
You now face south.
If you turn by a right angle in the anti-clockwise
direction, which direction will you face? It is east
again! (Why?)
Study the following positions :
Do This
Y ou stand facing
north
By a ‘right-angle-turn’
clockwise, you now
face east
By another
‘right-angle-turn’ you
finally face south.
Rationalised 2023-24
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
Fractions and Fractions and
Fractions and Fractions and Fractions and
Integers Integers
Integers Integers Integers
UNDERSTANDING ELEMENTARY SHAPES
63
From facing north to facing south, you have turned by
two right angles. Is not this the same as a single turn by
two right angles?
The turn from north to east is by a right angle.
The turn from north to south is by two right angles; it
is called a straight angle. (NS is a straight line!)
Stand facing south.
Turn by a straight angle.
Which direction do you face now?
You face north!
To turn from north to south, you took a straight angle
turn, again to turn from south to north, you took another
straight angle turn in the same direction. Thus, turning by
two straight angles you reach your original position.
Think, discuss and write
By how many right angles should you turn in the same direction to reach your
original position?
Turning by two straight angles (or four right angles) in the same direction
makes a full turn. This one complete turn is called one revolution. The angle
for one revolution is a complete angle.
We can see such revolutions on clock-faces. When the
hand of a clock moves from one position to another, it turns
through an angle.
Suppose the hand of a clock starts at 12 and goes round
until it reaches at 12 again. Has it not made one revolution?
So, how many right angles has it moved? Consider these
examples :
From 12 to 6 From 6 to 9 From 1 to 10
1
2
 of a revolution.
1
4
 of a revolution
3
4
 of a revolution
or 2 right angles. or 1 right angle. or 3 right angles.
Rationalised 2023-24