NCERT Textbook: Basic Geometrical Ideas | Mathematics (Maths) Class 6 PDF Download

 Page 1


Geometry has a long and rich history. The term ‘Geometry’ is the English
equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’
means Measurement. According to
historians, the geometrical ideas shaped up
in ancient times, probably due to the need
in art, architecture and measurement. These
include occasions when the boundaries of
cultivated lands had to be marked without
giving room for complaints. Construction of
magnificent palaces, temples, lakes, dams
and cities, art and architecture propped up
these ideas. Even today geometrical ideas
are reflected in all forms of art,
measurements, architecture, engineering, cloth designing etc. You observe
and use different objects like boxes, tables, books, the tiffin box you carry
to your school for lunch, the ball with which you play and
so on. All such objects have different shapes. The ruler which you use, the
pencil with which you write are straight. The pictures of a bangle, the one
rupee coin or a ball appear round.
Here, you will learn some interesting facts that will help you know more
about the shapes around you.
4.2 Points
By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner
will be the dot. This almost invisible tiny dot will give you an idea of a point.
4.1 Introduction
Chapter 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4
Basic Geometrical Basic Geometrical
Basic Geometrical Basic Geometrical Basic Geometrical
Ideas Ideas
Ideas Ideas Ideas
Rationalised 2023-24
Page 2


Geometry has a long and rich history. The term ‘Geometry’ is the English
equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’
means Measurement. According to
historians, the geometrical ideas shaped up
in ancient times, probably due to the need
in art, architecture and measurement. These
include occasions when the boundaries of
cultivated lands had to be marked without
giving room for complaints. Construction of
magnificent palaces, temples, lakes, dams
and cities, art and architecture propped up
these ideas. Even today geometrical ideas
are reflected in all forms of art,
measurements, architecture, engineering, cloth designing etc. You observe
and use different objects like boxes, tables, books, the tiffin box you carry
to your school for lunch, the ball with which you play and
so on. All such objects have different shapes. The ruler which you use, the
pencil with which you write are straight. The pictures of a bangle, the one
rupee coin or a ball appear round.
Here, you will learn some interesting facts that will help you know more
about the shapes around you.
4.2 Points
By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner
will be the dot. This almost invisible tiny dot will give you an idea of a point.
4.1 Introduction
Chapter 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4
Basic Geometrical Basic Geometrical
Basic Geometrical Basic Geometrical Basic Geometrical
Ideas Ideas
Ideas Ideas Ideas
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
47
A point determines
a location.
These are some
models for a point :
If you mark three
points on a paper, you
would be required to
distinguish them. For
this they are denoted
by a single capital letter like A,B,C.
These points will be read as point A, point B and point C.
Of course, the dots have to be invisibly thin.
1. With a sharp tip of the pencil, mark four points on a paper and name them
by the letters A,C,P ,H. Try to name these points in different ways. One such
way could be this
2. A star in the sky also gives us an idea of a point. Identify at least five such
situations in your daily life.
4.3 A Line Segment
Fold a piece of paper and unfold it. Do you see
a fold? This gives the idea of a line segment. It
has two end points A and B.
Take a thin thread. Hold its two ends and
stretch it without a slack. It represents a line
segment. The ends held by hands are the end
points of the line segment.
The tip of a
compass
The sharpened
end of a pencil
The pointed end of
a needle
lA
lB
lC
Rationalised 2023-24
Page 3


Geometry has a long and rich history. The term ‘Geometry’ is the English
equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’
means Measurement. According to
historians, the geometrical ideas shaped up
in ancient times, probably due to the need
in art, architecture and measurement. These
include occasions when the boundaries of
cultivated lands had to be marked without
giving room for complaints. Construction of
magnificent palaces, temples, lakes, dams
and cities, art and architecture propped up
these ideas. Even today geometrical ideas
are reflected in all forms of art,
measurements, architecture, engineering, cloth designing etc. You observe
and use different objects like boxes, tables, books, the tiffin box you carry
to your school for lunch, the ball with which you play and
so on. All such objects have different shapes. The ruler which you use, the
pencil with which you write are straight. The pictures of a bangle, the one
rupee coin or a ball appear round.
Here, you will learn some interesting facts that will help you know more
about the shapes around you.
4.2 Points
By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner
will be the dot. This almost invisible tiny dot will give you an idea of a point.
4.1 Introduction
Chapter 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4
Basic Geometrical Basic Geometrical
Basic Geometrical Basic Geometrical Basic Geometrical
Ideas Ideas
Ideas Ideas Ideas
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
47
A point determines
a location.
These are some
models for a point :
If you mark three
points on a paper, you
would be required to
distinguish them. For
this they are denoted
by a single capital letter like A,B,C.
These points will be read as point A, point B and point C.
Of course, the dots have to be invisibly thin.
1. With a sharp tip of the pencil, mark four points on a paper and name them
by the letters A,C,P ,H. Try to name these points in different ways. One such
way could be this
2. A star in the sky also gives us an idea of a point. Identify at least five such
situations in your daily life.
4.3 A Line Segment
Fold a piece of paper and unfold it. Do you see
a fold? This gives the idea of a line segment. It
has two end points A and B.
Take a thin thread. Hold its two ends and
stretch it without a slack. It represents a line
segment. The ends held by hands are the end
points of the line segment.
The tip of a
compass
The sharpened
end of a pencil
The pointed end of
a needle
lA
lB
lC
Rationalised 2023-24
MATHEMATICS
48
The following are some models for a line segment :
Try to find more examples for line segments
from your surroundings.
Mark any two points A and B on a sheet
of paper. Try to connect A to B by all possible
routes. (Fig 4.1)
What is the shortest route from A to B?
This shortest join of point A to B
(including A and B) shown here is a line
segment. It is denoted by  or  . The points A and B are called the end
points of the segment.
1. Name the line segments in the figure 4.2.
Is A, the end point of each line segment?
An edge of
a box A tube light
A
B
Fig 4.2
4.4 A Line
Imagine that the line segment from A to B (i.e. AB ) is extended beyond A in
one direction and beyond B in the other direction
without any end (see figure). You now get a model
for a line.
Do you think you can draw a complete picture of a line? No. (Why?)
A line through two points A and B is written as 
AB
s r uu
. It extends
indefinitely in both directions. So it contains a countless
number of points. (Think about this).
Two points are enough to fix a line. We say ‘two points
determine a line’.
The adjacent diagram (Fig 4.3) is that of a line PQ written
as PQ
s r uu
. Sometimes a line is denoted by a letter like l, m.
Fig 4.1
Fig 4.3
The edge of a post card
Rationalised 2023-24
Page 4


Geometry has a long and rich history. The term ‘Geometry’ is the English
equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’
means Measurement. According to
historians, the geometrical ideas shaped up
in ancient times, probably due to the need
in art, architecture and measurement. These
include occasions when the boundaries of
cultivated lands had to be marked without
giving room for complaints. Construction of
magnificent palaces, temples, lakes, dams
and cities, art and architecture propped up
these ideas. Even today geometrical ideas
are reflected in all forms of art,
measurements, architecture, engineering, cloth designing etc. You observe
and use different objects like boxes, tables, books, the tiffin box you carry
to your school for lunch, the ball with which you play and
so on. All such objects have different shapes. The ruler which you use, the
pencil with which you write are straight. The pictures of a bangle, the one
rupee coin or a ball appear round.
Here, you will learn some interesting facts that will help you know more
about the shapes around you.
4.2 Points
By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner
will be the dot. This almost invisible tiny dot will give you an idea of a point.
4.1 Introduction
Chapter 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4
Basic Geometrical Basic Geometrical
Basic Geometrical Basic Geometrical Basic Geometrical
Ideas Ideas
Ideas Ideas Ideas
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
47
A point determines
a location.
These are some
models for a point :
If you mark three
points on a paper, you
would be required to
distinguish them. For
this they are denoted
by a single capital letter like A,B,C.
These points will be read as point A, point B and point C.
Of course, the dots have to be invisibly thin.
1. With a sharp tip of the pencil, mark four points on a paper and name them
by the letters A,C,P ,H. Try to name these points in different ways. One such
way could be this
2. A star in the sky also gives us an idea of a point. Identify at least five such
situations in your daily life.
4.3 A Line Segment
Fold a piece of paper and unfold it. Do you see
a fold? This gives the idea of a line segment. It
has two end points A and B.
Take a thin thread. Hold its two ends and
stretch it without a slack. It represents a line
segment. The ends held by hands are the end
points of the line segment.
The tip of a
compass
The sharpened
end of a pencil
The pointed end of
a needle
lA
lB
lC
Rationalised 2023-24
MATHEMATICS
48
The following are some models for a line segment :
Try to find more examples for line segments
from your surroundings.
Mark any two points A and B on a sheet
of paper. Try to connect A to B by all possible
routes. (Fig 4.1)
What is the shortest route from A to B?
This shortest join of point A to B
(including A and B) shown here is a line
segment. It is denoted by  or  . The points A and B are called the end
points of the segment.
1. Name the line segments in the figure 4.2.
Is A, the end point of each line segment?
An edge of
a box A tube light
A
B
Fig 4.2
4.4 A Line
Imagine that the line segment from A to B (i.e. AB ) is extended beyond A in
one direction and beyond B in the other direction
without any end (see figure). You now get a model
for a line.
Do you think you can draw a complete picture of a line? No. (Why?)
A line through two points A and B is written as 
AB
s r uu
. It extends
indefinitely in both directions. So it contains a countless
number of points. (Think about this).
Two points are enough to fix a line. We say ‘two points
determine a line’.
The adjacent diagram (Fig 4.3) is that of a line PQ written
as PQ
s r uu
. Sometimes a line is denoted by a letter like l, m.
Fig 4.1
Fig 4.3
The edge of a post card
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
49
4.5 Intersecting Lines
Look at the diagram (Fig 4.4). Two lines l
1
 and l
2
are shown. Both the lines pass through point P.
We say l
1
 and l
2
 intersect at P. If two lines have
one  common point, they are called intersecting
lines.
The following are some models of a pair of
intersecting lines (Fig 4.5) :
Try to find out some more models for a pair of intersecting lines.
Take a sheet of paper. Make two folds (and crease them) to represent a pair of
intersecting lines and discuss :
(a) Can two lines intersect in more than one point?
(b) Can more than two lines intersect in one point?
4.6 Parallel Lines
Let us look at this table (Fig 4.6). The top ABCD is flat. Are you able to see
some points and line segments?
Are there intersecting line segments?
Yes, AB
    
 and BC
    
 intersect at the
point B.
Which line segments intersect at A?
at C? at D?
Do the lines AD
    
 and CD
    
 intersect?
Fig 4.4
Two adjacement edges
of your notebook
The letter X of the
English alphabet
Crossing-roads
Fig 4.5
Fig 4.6
Rationalised 2023-24
Page 5


Geometry has a long and rich history. The term ‘Geometry’ is the English
equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’
means Measurement. According to
historians, the geometrical ideas shaped up
in ancient times, probably due to the need
in art, architecture and measurement. These
include occasions when the boundaries of
cultivated lands had to be marked without
giving room for complaints. Construction of
magnificent palaces, temples, lakes, dams
and cities, art and architecture propped up
these ideas. Even today geometrical ideas
are reflected in all forms of art,
measurements, architecture, engineering, cloth designing etc. You observe
and use different objects like boxes, tables, books, the tiffin box you carry
to your school for lunch, the ball with which you play and
so on. All such objects have different shapes. The ruler which you use, the
pencil with which you write are straight. The pictures of a bangle, the one
rupee coin or a ball appear round.
Here, you will learn some interesting facts that will help you know more
about the shapes around you.
4.2 Points
By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner
will be the dot. This almost invisible tiny dot will give you an idea of a point.
4.1 Introduction
Chapter 4 Chapter 4 Chapter 4 Chapter 4 Chapter 4
Basic Geometrical Basic Geometrical
Basic Geometrical Basic Geometrical Basic Geometrical
Ideas Ideas
Ideas Ideas Ideas
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
47
A point determines
a location.
These are some
models for a point :
If you mark three
points on a paper, you
would be required to
distinguish them. For
this they are denoted
by a single capital letter like A,B,C.
These points will be read as point A, point B and point C.
Of course, the dots have to be invisibly thin.
1. With a sharp tip of the pencil, mark four points on a paper and name them
by the letters A,C,P ,H. Try to name these points in different ways. One such
way could be this
2. A star in the sky also gives us an idea of a point. Identify at least five such
situations in your daily life.
4.3 A Line Segment
Fold a piece of paper and unfold it. Do you see
a fold? This gives the idea of a line segment. It
has two end points A and B.
Take a thin thread. Hold its two ends and
stretch it without a slack. It represents a line
segment. The ends held by hands are the end
points of the line segment.
The tip of a
compass
The sharpened
end of a pencil
The pointed end of
a needle
lA
lB
lC
Rationalised 2023-24
MATHEMATICS
48
The following are some models for a line segment :
Try to find more examples for line segments
from your surroundings.
Mark any two points A and B on a sheet
of paper. Try to connect A to B by all possible
routes. (Fig 4.1)
What is the shortest route from A to B?
This shortest join of point A to B
(including A and B) shown here is a line
segment. It is denoted by  or  . The points A and B are called the end
points of the segment.
1. Name the line segments in the figure 4.2.
Is A, the end point of each line segment?
An edge of
a box A tube light
A
B
Fig 4.2
4.4 A Line
Imagine that the line segment from A to B (i.e. AB ) is extended beyond A in
one direction and beyond B in the other direction
without any end (see figure). You now get a model
for a line.
Do you think you can draw a complete picture of a line? No. (Why?)
A line through two points A and B is written as 
AB
s r uu
. It extends
indefinitely in both directions. So it contains a countless
number of points. (Think about this).
Two points are enough to fix a line. We say ‘two points
determine a line’.
The adjacent diagram (Fig 4.3) is that of a line PQ written
as PQ
s r uu
. Sometimes a line is denoted by a letter like l, m.
Fig 4.1
Fig 4.3
The edge of a post card
Rationalised 2023-24
BASIC GEOMETRICAL IDEAS
49
4.5 Intersecting Lines
Look at the diagram (Fig 4.4). Two lines l
1
 and l
2
are shown. Both the lines pass through point P.
We say l
1
 and l
2
 intersect at P. If two lines have
one  common point, they are called intersecting
lines.
The following are some models of a pair of
intersecting lines (Fig 4.5) :
Try to find out some more models for a pair of intersecting lines.
Take a sheet of paper. Make two folds (and crease them) to represent a pair of
intersecting lines and discuss :
(a) Can two lines intersect in more than one point?
(b) Can more than two lines intersect in one point?
4.6 Parallel Lines
Let us look at this table (Fig 4.6). The top ABCD is flat. Are you able to see
some points and line segments?
Are there intersecting line segments?
Yes, AB
    
 and BC
    
 intersect at the
point B.
Which line segments intersect at A?
at C? at D?
Do the lines AD
    
 and CD
    
 intersect?
Fig 4.4
Two adjacement edges
of your notebook
The letter X of the
English alphabet
Crossing-roads
Fig 4.5
Fig 4.6
Rationalised 2023-24
MATHEMATICS
50
Do the lines 
AD
s r uu
 and BC
s r uu
 intersect?
You find that on the table’s surface there are line segment which will not
meet, however far they are extended. 
AD
s r uu
 and BC
s r uu
 form one such pair. Can
you identify one more such pair of lines (which do not meet) on the top of
the table?
Lines like these which do not meet are said to be parallel; and are called
parallel lines.
Think, discuss and write
Where else do you see parallel lines? Try to find ten examples.
If two lines 
AB
s r uu
 and 
CD
s r uu
 are parallel, we write 
AB
s r uu
 || 
CD
s r uu
.
If two lines l
1
 and l
2
 are parallel, we write l
1
 || l
2
 .
Can you identify parrallel lines in the following figures?
Ray of light
from a torch Sun rays
Beam of light from
a light house
4.7 Ray
Rail lines
The opposite edges of ruler (scale) The cross-bars of this window
Rationalised 2023-24