NCERT Textbook: Practical Geometry | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


We see a number of shapes with which we are familiar. We also make a lot of
pictures. These pictures include different shapes. We have learnt about some of
these shapes in earlier chapters as well. Why don’t you list those shapes that
you know about alongwith how they appear?
In this chapter we shall learn to make these shapes. In making these shapes
we need to use some tools. We shall begin with listing these tools, describing
them and looking at how they are used.
S.No. Name and figure Description Use
1. The  Ruler A ruler ideally has no To draw line
[or the straight markings on it.  However, segments and
edge] the ruler in your instruments to measure
box is graduated into their lengths.
centimetres along one edge
(and sometimes into inches
along the other edge).
2. The Compasses A pair – a pointer on one To mark off
end and a pencil on the equal lengths
other. but not to
measure them.
To draw arcs
and circles.
Pencil Pointer
14.1 Introduction
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
Practical Practical
Practical Practical Practical
Geometry Geometry
Geometry Geometry Geometry
2022-23
Page 2


We see a number of shapes with which we are familiar. We also make a lot of
pictures. These pictures include different shapes. We have learnt about some of
these shapes in earlier chapters as well. Why don’t you list those shapes that
you know about alongwith how they appear?
In this chapter we shall learn to make these shapes. In making these shapes
we need to use some tools. We shall begin with listing these tools, describing
them and looking at how they are used.
S.No. Name and figure Description Use
1. The  Ruler A ruler ideally has no To draw line
[or the straight markings on it.  However, segments and
edge] the ruler in your instruments to measure
box is graduated into their lengths.
centimetres along one edge
(and sometimes into inches
along the other edge).
2. The Compasses A pair – a pointer on one To mark off
end and a pencil on the equal lengths
other. but not to
measure them.
To draw arcs
and circles.
Pencil Pointer
14.1 Introduction
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
Practical Practical
Practical Practical Practical
Geometry Geometry
Geometry Geometry Geometry
2022-23
PRACTICAL GEOMETRY
275
3. The Divider A pair of pointers To compare
lengths.
4. Set-Squares Two triangular To draw
pieces – one of them perpendicular
has 45°, 45°, 90° and parallel
angles at the vertices lines.
and the other has
30°, 60°, 90° angles
at the vertices.
5. The Protractor A semi-circular To draw
device graduated into and measure angles.
180 degree-parts.
The measure starts
from 0° on the right
hand side and ends
with 180° on the left
hand side and vice-versa.
We are going to consider “Ruler and compasses constructions”, using
ruler, only to draw lines, and compasses, only to draw arcs.
Be careful while doing these constructions.
Here are some tips to help you.
(a) Draw thin lines and mark points lightly.
(b) Maintain instruments with sharp tips and fine edges.
(c) Have two pencils in the box, one for insertion into the compasses and
the other to draw lines or curves and mark points.
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10 11 12 13
9 8 7 6 5 4 3 2 1
2022-23
Page 3


We see a number of shapes with which we are familiar. We also make a lot of
pictures. These pictures include different shapes. We have learnt about some of
these shapes in earlier chapters as well. Why don’t you list those shapes that
you know about alongwith how they appear?
In this chapter we shall learn to make these shapes. In making these shapes
we need to use some tools. We shall begin with listing these tools, describing
them and looking at how they are used.
S.No. Name and figure Description Use
1. The  Ruler A ruler ideally has no To draw line
[or the straight markings on it.  However, segments and
edge] the ruler in your instruments to measure
box is graduated into their lengths.
centimetres along one edge
(and sometimes into inches
along the other edge).
2. The Compasses A pair – a pointer on one To mark off
end and a pencil on the equal lengths
other. but not to
measure them.
To draw arcs
and circles.
Pencil Pointer
14.1 Introduction
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
Practical Practical
Practical Practical Practical
Geometry Geometry
Geometry Geometry Geometry
2022-23
PRACTICAL GEOMETRY
275
3. The Divider A pair of pointers To compare
lengths.
4. Set-Squares Two triangular To draw
pieces – one of them perpendicular
has 45°, 45°, 90° and parallel
angles at the vertices lines.
and the other has
30°, 60°, 90° angles
at the vertices.
5. The Protractor A semi-circular To draw
device graduated into and measure angles.
180 degree-parts.
The measure starts
from 0° on the right
hand side and ends
with 180° on the left
hand side and vice-versa.
We are going to consider “Ruler and compasses constructions”, using
ruler, only to draw lines, and compasses, only to draw arcs.
Be careful while doing these constructions.
Here are some tips to help you.
(a) Draw thin lines and mark points lightly.
(b) Maintain instruments with sharp tips and fine edges.
(c) Have two pencils in the box, one for insertion into the compasses and
the other to draw lines or curves and mark points.
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10 11 12 13
9 8 7 6 5 4 3 2 1
2022-23
MATHEMATICS
276
14.2 The Circle
Look at the wheel shown here. Every point on its boundary is
at an equal distance from its centre. Can you mention a few
such objects and draw them? Think about five such objects
which have this shape.
14.2.1  Construction of a circle when its radius is known
Suppose we want to draw a circle of radius 3 cm. W e need to use our compasses.
Here are the steps to follow.
Step 1Open the
compasses for the
required radius of 3cm.
Step 2 Mark a point
with a sharp pencil
where we want the
centre of the circle to
be. Name it as O.
Step 3 Place the pointer of the compasses on O.
Step 4 Turn the compasses slowly to draw the circle. Be careful to complete
the movement around in one instant.
Think, discuss and write
How many circles can you draw with a given centre O and a point, say P?
EXERCISE 14.1
1. Draw a circle of radius 3.2 cm.
2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm.
3. Draw a circle and any two of its diameters.  If you join the ends of these diameters,
what is the figure obtained? What figure is obtained if the diameters are
perpendicular to each other? How do you check your answer?
4. Draw any circle and mark points A, B and C such that
(a) A is on the circle. (b) B is in the interior of the circle.
(c) C is in the exterior of the circle.
5. Let A, B be the centres of two circles of equal radii; draw them so that each one
of them passes through the centre of the other. Let them intersect at C and D.
Examine whether AB and CD are at right angles.
2022-23
Page 4


We see a number of shapes with which we are familiar. We also make a lot of
pictures. These pictures include different shapes. We have learnt about some of
these shapes in earlier chapters as well. Why don’t you list those shapes that
you know about alongwith how they appear?
In this chapter we shall learn to make these shapes. In making these shapes
we need to use some tools. We shall begin with listing these tools, describing
them and looking at how they are used.
S.No. Name and figure Description Use
1. The  Ruler A ruler ideally has no To draw line
[or the straight markings on it.  However, segments and
edge] the ruler in your instruments to measure
box is graduated into their lengths.
centimetres along one edge
(and sometimes into inches
along the other edge).
2. The Compasses A pair – a pointer on one To mark off
end and a pencil on the equal lengths
other. but not to
measure them.
To draw arcs
and circles.
Pencil Pointer
14.1 Introduction
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
Practical Practical
Practical Practical Practical
Geometry Geometry
Geometry Geometry Geometry
2022-23
PRACTICAL GEOMETRY
275
3. The Divider A pair of pointers To compare
lengths.
4. Set-Squares Two triangular To draw
pieces – one of them perpendicular
has 45°, 45°, 90° and parallel
angles at the vertices lines.
and the other has
30°, 60°, 90° angles
at the vertices.
5. The Protractor A semi-circular To draw
device graduated into and measure angles.
180 degree-parts.
The measure starts
from 0° on the right
hand side and ends
with 180° on the left
hand side and vice-versa.
We are going to consider “Ruler and compasses constructions”, using
ruler, only to draw lines, and compasses, only to draw arcs.
Be careful while doing these constructions.
Here are some tips to help you.
(a) Draw thin lines and mark points lightly.
(b) Maintain instruments with sharp tips and fine edges.
(c) Have two pencils in the box, one for insertion into the compasses and
the other to draw lines or curves and mark points.
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10 11 12 13
9 8 7 6 5 4 3 2 1
2022-23
MATHEMATICS
276
14.2 The Circle
Look at the wheel shown here. Every point on its boundary is
at an equal distance from its centre. Can you mention a few
such objects and draw them? Think about five such objects
which have this shape.
14.2.1  Construction of a circle when its radius is known
Suppose we want to draw a circle of radius 3 cm. W e need to use our compasses.
Here are the steps to follow.
Step 1Open the
compasses for the
required radius of 3cm.
Step 2 Mark a point
with a sharp pencil
where we want the
centre of the circle to
be. Name it as O.
Step 3 Place the pointer of the compasses on O.
Step 4 Turn the compasses slowly to draw the circle. Be careful to complete
the movement around in one instant.
Think, discuss and write
How many circles can you draw with a given centre O and a point, say P?
EXERCISE 14.1
1. Draw a circle of radius 3.2 cm.
2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm.
3. Draw a circle and any two of its diameters.  If you join the ends of these diameters,
what is the figure obtained? What figure is obtained if the diameters are
perpendicular to each other? How do you check your answer?
4. Draw any circle and mark points A, B and C such that
(a) A is on the circle. (b) B is in the interior of the circle.
(c) C is in the exterior of the circle.
5. Let A, B be the centres of two circles of equal radii; draw them so that each one
of them passes through the centre of the other. Let them intersect at C and D.
Examine whether AB and CD are at right angles.
2022-23
PRACTICAL GEOMETRY
277
14.3 A Line Segment
Remember that a line segment has two end points. This makes it possible to
measure its length with a ruler.
If we know the length of a line segment, it becomes possible to represent
it by a diagram. Let us see how we do this.
14.3.1 Construction of a line segment of a given length
Suppose we want to draw a line segment of length 4.7 cm. We can use our
ruler and mark two points A and B which are 4.7 cm apart.  Join A and B and
get AB . While marking the points A and B, we should look straight down at
the measuring device. Otherwise we will get an incorrect value.
Use of ruler and compasses
A better method would be to use compasses to construct a line segment of a
given length.
Step 1 Draw a line l. Mark a point A on a line l.
Step 2 Place the compasses pointer on the zero mark
of the ruler. Open it to place the pencil point upto
the 4.7cm mark.
Step 3 Taking caution that the opening of the
compasses has not changed, place the pointer on
A and swing an arc to cut l at B.
Step 4 AB is a line segment of required length.
l
2022-23
Page 5


We see a number of shapes with which we are familiar. We also make a lot of
pictures. These pictures include different shapes. We have learnt about some of
these shapes in earlier chapters as well. Why don’t you list those shapes that
you know about alongwith how they appear?
In this chapter we shall learn to make these shapes. In making these shapes
we need to use some tools. We shall begin with listing these tools, describing
them and looking at how they are used.
S.No. Name and figure Description Use
1. The  Ruler A ruler ideally has no To draw line
[or the straight markings on it.  However, segments and
edge] the ruler in your instruments to measure
box is graduated into their lengths.
centimetres along one edge
(and sometimes into inches
along the other edge).
2. The Compasses A pair – a pointer on one To mark off
end and a pencil on the equal lengths
other. but not to
measure them.
To draw arcs
and circles.
Pencil Pointer
14.1 Introduction
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
Practical Practical
Practical Practical Practical
Geometry Geometry
Geometry Geometry Geometry
2022-23
PRACTICAL GEOMETRY
275
3. The Divider A pair of pointers To compare
lengths.
4. Set-Squares Two triangular To draw
pieces – one of them perpendicular
has 45°, 45°, 90° and parallel
angles at the vertices lines.
and the other has
30°, 60°, 90° angles
at the vertices.
5. The Protractor A semi-circular To draw
device graduated into and measure angles.
180 degree-parts.
The measure starts
from 0° on the right
hand side and ends
with 180° on the left
hand side and vice-versa.
We are going to consider “Ruler and compasses constructions”, using
ruler, only to draw lines, and compasses, only to draw arcs.
Be careful while doing these constructions.
Here are some tips to help you.
(a) Draw thin lines and mark points lightly.
(b) Maintain instruments with sharp tips and fine edges.
(c) Have two pencils in the box, one for insertion into the compasses and
the other to draw lines or curves and mark points.
1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10 11 12 13
9 8 7 6 5 4 3 2 1
2022-23
MATHEMATICS
276
14.2 The Circle
Look at the wheel shown here. Every point on its boundary is
at an equal distance from its centre. Can you mention a few
such objects and draw them? Think about five such objects
which have this shape.
14.2.1  Construction of a circle when its radius is known
Suppose we want to draw a circle of radius 3 cm. W e need to use our compasses.
Here are the steps to follow.
Step 1Open the
compasses for the
required radius of 3cm.
Step 2 Mark a point
with a sharp pencil
where we want the
centre of the circle to
be. Name it as O.
Step 3 Place the pointer of the compasses on O.
Step 4 Turn the compasses slowly to draw the circle. Be careful to complete
the movement around in one instant.
Think, discuss and write
How many circles can you draw with a given centre O and a point, say P?
EXERCISE 14.1
1. Draw a circle of radius 3.2 cm.
2. With the same centre O, draw two circles of radii 4 cm and 2.5 cm.
3. Draw a circle and any two of its diameters.  If you join the ends of these diameters,
what is the figure obtained? What figure is obtained if the diameters are
perpendicular to each other? How do you check your answer?
4. Draw any circle and mark points A, B and C such that
(a) A is on the circle. (b) B is in the interior of the circle.
(c) C is in the exterior of the circle.
5. Let A, B be the centres of two circles of equal radii; draw them so that each one
of them passes through the centre of the other. Let them intersect at C and D.
Examine whether AB and CD are at right angles.
2022-23
PRACTICAL GEOMETRY
277
14.3 A Line Segment
Remember that a line segment has two end points. This makes it possible to
measure its length with a ruler.
If we know the length of a line segment, it becomes possible to represent
it by a diagram. Let us see how we do this.
14.3.1 Construction of a line segment of a given length
Suppose we want to draw a line segment of length 4.7 cm. We can use our
ruler and mark two points A and B which are 4.7 cm apart.  Join A and B and
get AB . While marking the points A and B, we should look straight down at
the measuring device. Otherwise we will get an incorrect value.
Use of ruler and compasses
A better method would be to use compasses to construct a line segment of a
given length.
Step 1 Draw a line l. Mark a point A on a line l.
Step 2 Place the compasses pointer on the zero mark
of the ruler. Open it to place the pencil point upto
the 4.7cm mark.
Step 3 Taking caution that the opening of the
compasses has not changed, place the pointer on
A and swing an arc to cut l at B.
Step 4 AB is a line segment of required length.
l
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MATHEMATICS
278
EXERCISE 14.2
1. Draw a line segment of length 7.3 cm using a ruler.
2. Construct a line segment of length 5.6 cm using ruler and compasses.
3. Construct AB of length 7.8 cm.  From this, cut off AC of length 4.7 cm. Measure
BC .
4. Given AB of length 3.9 cm,  construct PQ such that the length of PQ is twice
that of AB . V erify by measurement.
(Hint : Construct PX such that length of PX = length of AB ;
then cut off 
XQ
 such that 
XQ
 also has the length of AB .)
5. Given AB of length 7.3 cm and CD of length 3.4 cm,  construct
a line segment XY such that the length of XY is equal to  the difference between
the lengths of AB and CD . V erify by measurement.
14.3.2 Constructing a copy of a given line segment
Suppose you want to draw a line segment whose length is equal to that of a
given line segment AB .
A quick and natural approach is to use your ruler (which is marked with
centimetres and millimetres) to measure the length of AB and then use the
same length to draw another line segment CD .
A second approach would be to use a transparent sheet and trace  AB onto
another portion of the paper. But these methods may not always give accurate
results.
A better approach would be to use ruler and compasses for making this
construction.
To make a copy of AB .
Step 1 Given AB whose length is not known.
A B
2022-23