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

 Page 1


MATHEMATICS 144
9.1  AREA OF A PARALLELOGRAM
We come across many shapes other than squares and rectangles.
How will you find the area of a land which is a parallelogram in shape?
Let us find a method to get the area of a parallelogram.
Can a parallelogram be converted into a rectangle of equal area?
Draw a parallelogram on a graph paper as shown in Fig 9.1(i). Cut out the
parallelogram. Draw a line from one vertex of the parallelogram perpendicular to the
opposite side [Fig 9.1(ii)]. Cut out the triangle. Move the triangle to the other side of the
parallelogram.
Perimeter and
Area Chapter  9
(i) (ii) (iii)
Fig 9.1
Fig 9.2
What shape do you get? Y ou get a rectangle.
Is the area of the parallelogram equal to the area
of the rectangle formed?
Yes, area of the parallelogram = area of the
rectangle formed
What are the length and the breadth of the
rectangle?
2024-25
Page 2


MATHEMATICS 144
9.1  AREA OF A PARALLELOGRAM
We come across many shapes other than squares and rectangles.
How will you find the area of a land which is a parallelogram in shape?
Let us find a method to get the area of a parallelogram.
Can a parallelogram be converted into a rectangle of equal area?
Draw a parallelogram on a graph paper as shown in Fig 9.1(i). Cut out the
parallelogram. Draw a line from one vertex of the parallelogram perpendicular to the
opposite side [Fig 9.1(ii)]. Cut out the triangle. Move the triangle to the other side of the
parallelogram.
Perimeter and
Area Chapter  9
(i) (ii) (iii)
Fig 9.1
Fig 9.2
What shape do you get? Y ou get a rectangle.
Is the area of the parallelogram equal to the area
of the rectangle formed?
Yes, area of the parallelogram = area of the
rectangle formed
What are the length and the breadth of the
rectangle?
2024-25
PERIMETER AND AREA 145
W e find that the length of the rectangle formed is equal to the base of the parallelogram
and the breadth of the rectangle is equal to the height of the parallelogram (Fig 9.2).
Now, Area of parallelogram = Area of rectangle
= length × breadth = l × b
But the length l and breadth b of the  rectangle are exactly the
base b and the height h, respectively of the parallelogram.
Thus, the area of parallelogram = base × height = b × h.
D C
A
E
base
B
height
Any side of a parallelogram can be chosen as base of the
parallelogram. The perpendicular dropped on that side from the opposite
vertex is known as height (altitude). In the parallelogram ABCD, DE is
perpendicular to AB.  Here AB is the
base and DE is the height of the
parallelogram.
In this parallelogram ABCD, BF is the
perpendicular to opposite side AD. Here AD is the
base and BF is the height.
base
D
C
A
B
height
F
Consider the following parallelograms (Fig 9.2).
Fig 9.3
Find the areas of the parallelograms by counting the squares enclosed within the figures
and also find the perimeters by measuring the sides.
2024-25
Page 3


MATHEMATICS 144
9.1  AREA OF A PARALLELOGRAM
We come across many shapes other than squares and rectangles.
How will you find the area of a land which is a parallelogram in shape?
Let us find a method to get the area of a parallelogram.
Can a parallelogram be converted into a rectangle of equal area?
Draw a parallelogram on a graph paper as shown in Fig 9.1(i). Cut out the
parallelogram. Draw a line from one vertex of the parallelogram perpendicular to the
opposite side [Fig 9.1(ii)]. Cut out the triangle. Move the triangle to the other side of the
parallelogram.
Perimeter and
Area Chapter  9
(i) (ii) (iii)
Fig 9.1
Fig 9.2
What shape do you get? Y ou get a rectangle.
Is the area of the parallelogram equal to the area
of the rectangle formed?
Yes, area of the parallelogram = area of the
rectangle formed
What are the length and the breadth of the
rectangle?
2024-25
PERIMETER AND AREA 145
W e find that the length of the rectangle formed is equal to the base of the parallelogram
and the breadth of the rectangle is equal to the height of the parallelogram (Fig 9.2).
Now, Area of parallelogram = Area of rectangle
= length × breadth = l × b
But the length l and breadth b of the  rectangle are exactly the
base b and the height h, respectively of the parallelogram.
Thus, the area of parallelogram = base × height = b × h.
D C
A
E
base
B
height
Any side of a parallelogram can be chosen as base of the
parallelogram. The perpendicular dropped on that side from the opposite
vertex is known as height (altitude). In the parallelogram ABCD, DE is
perpendicular to AB.  Here AB is the
base and DE is the height of the
parallelogram.
In this parallelogram ABCD, BF is the
perpendicular to opposite side AD. Here AD is the
base and BF is the height.
base
D
C
A
B
height
F
Consider the following parallelograms (Fig 9.2).
Fig 9.3
Find the areas of the parallelograms by counting the squares enclosed within the figures
and also find the perimeters by measuring the sides.
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MATHEMATICS 146
Complete the following table:
Parallelogram Base Height Area Perimeter
(a) 5 units 3 units 15 sq units
(b)
(c)
(d)
(e)
(f)
(g)
Y ou will find that all these parallelograms have equal areas but different perimeters. Now ,
consider the following parallelograms with sides 7 cm and 5 cm (Fig 9.4).
Fig 9.4
Find the perimeter and area of each of these parallelograms. Analyse your results.
Y ou will find that these parallelograms have different areas but equal perimeters.
To find the area of a parallelogram, you need to know only the base and the
corresponding height of the parallelogram.
2024-25
Page 4


MATHEMATICS 144
9.1  AREA OF A PARALLELOGRAM
We come across many shapes other than squares and rectangles.
How will you find the area of a land which is a parallelogram in shape?
Let us find a method to get the area of a parallelogram.
Can a parallelogram be converted into a rectangle of equal area?
Draw a parallelogram on a graph paper as shown in Fig 9.1(i). Cut out the
parallelogram. Draw a line from one vertex of the parallelogram perpendicular to the
opposite side [Fig 9.1(ii)]. Cut out the triangle. Move the triangle to the other side of the
parallelogram.
Perimeter and
Area Chapter  9
(i) (ii) (iii)
Fig 9.1
Fig 9.2
What shape do you get? Y ou get a rectangle.
Is the area of the parallelogram equal to the area
of the rectangle formed?
Yes, area of the parallelogram = area of the
rectangle formed
What are the length and the breadth of the
rectangle?
2024-25
PERIMETER AND AREA 145
W e find that the length of the rectangle formed is equal to the base of the parallelogram
and the breadth of the rectangle is equal to the height of the parallelogram (Fig 9.2).
Now, Area of parallelogram = Area of rectangle
= length × breadth = l × b
But the length l and breadth b of the  rectangle are exactly the
base b and the height h, respectively of the parallelogram.
Thus, the area of parallelogram = base × height = b × h.
D C
A
E
base
B
height
Any side of a parallelogram can be chosen as base of the
parallelogram. The perpendicular dropped on that side from the opposite
vertex is known as height (altitude). In the parallelogram ABCD, DE is
perpendicular to AB.  Here AB is the
base and DE is the height of the
parallelogram.
In this parallelogram ABCD, BF is the
perpendicular to opposite side AD. Here AD is the
base and BF is the height.
base
D
C
A
B
height
F
Consider the following parallelograms (Fig 9.2).
Fig 9.3
Find the areas of the parallelograms by counting the squares enclosed within the figures
and also find the perimeters by measuring the sides.
2024-25
MATHEMATICS 146
Complete the following table:
Parallelogram Base Height Area Perimeter
(a) 5 units 3 units 15 sq units
(b)
(c)
(d)
(e)
(f)
(g)
Y ou will find that all these parallelograms have equal areas but different perimeters. Now ,
consider the following parallelograms with sides 7 cm and 5 cm (Fig 9.4).
Fig 9.4
Find the perimeter and area of each of these parallelograms. Analyse your results.
Y ou will find that these parallelograms have different areas but equal perimeters.
To find the area of a parallelogram, you need to know only the base and the
corresponding height of the parallelogram.
2024-25
PERIMETER AND AREA 147
TRY THESE
Find the area of following parallelograms:
(i) (ii)
(iii) In a parallelogram ABCD, AB = 7.2 cm and the perpendicular from C on AB is 4.5 cm.
9.2  AREA OF A TRIANGLE
A gardener wants to know the cost of covering the whole of a triangular
garden with grass.
In this case we need to know the area of the triangular region.
Let us find a method to get the area of a triangle.
Draw a scalene triangle on a piece of paper. Cut out the triangle.
Place this triangle on another piece of paper and cut out another
triangle of the same size.
So now you have two scalene triangles of the same size.
Are both the triangles congruent?
Superpose one triangle on the other so that they match.
Y ou may have to rotate one of the two triangles.
Now place both the triangles such that a pair of corresponding
sides is joined as shown in Fig 9.5.
Is the figure thus formed a parallelogram?
Compare the area of each triangle to the area of the
parallelogram.
Compare the base and height of the triangles with the base
and height of the parallelogram.
Y ou will find that the sum of the areas of both the triangles is
equal to the area of the parallelogram. The base and the height
of the triangle are the same as the base and the height of the
parallelogram, respectively .
Area of each triangle =
1
2
(Area of parallelogram)
=
1
2
(base × height) (Since area of a parallelogram = base × height)
=
1
2
( ) b h ×
 (or 
1
2
bh
, in short)
Fig 9.5
D
E
F
A
B
C
2024-25
Page 5


MATHEMATICS 144
9.1  AREA OF A PARALLELOGRAM
We come across many shapes other than squares and rectangles.
How will you find the area of a land which is a parallelogram in shape?
Let us find a method to get the area of a parallelogram.
Can a parallelogram be converted into a rectangle of equal area?
Draw a parallelogram on a graph paper as shown in Fig 9.1(i). Cut out the
parallelogram. Draw a line from one vertex of the parallelogram perpendicular to the
opposite side [Fig 9.1(ii)]. Cut out the triangle. Move the triangle to the other side of the
parallelogram.
Perimeter and
Area Chapter  9
(i) (ii) (iii)
Fig 9.1
Fig 9.2
What shape do you get? Y ou get a rectangle.
Is the area of the parallelogram equal to the area
of the rectangle formed?
Yes, area of the parallelogram = area of the
rectangle formed
What are the length and the breadth of the
rectangle?
2024-25
PERIMETER AND AREA 145
W e find that the length of the rectangle formed is equal to the base of the parallelogram
and the breadth of the rectangle is equal to the height of the parallelogram (Fig 9.2).
Now, Area of parallelogram = Area of rectangle
= length × breadth = l × b
But the length l and breadth b of the  rectangle are exactly the
base b and the height h, respectively of the parallelogram.
Thus, the area of parallelogram = base × height = b × h.
D C
A
E
base
B
height
Any side of a parallelogram can be chosen as base of the
parallelogram. The perpendicular dropped on that side from the opposite
vertex is known as height (altitude). In the parallelogram ABCD, DE is
perpendicular to AB.  Here AB is the
base and DE is the height of the
parallelogram.
In this parallelogram ABCD, BF is the
perpendicular to opposite side AD. Here AD is the
base and BF is the height.
base
D
C
A
B
height
F
Consider the following parallelograms (Fig 9.2).
Fig 9.3
Find the areas of the parallelograms by counting the squares enclosed within the figures
and also find the perimeters by measuring the sides.
2024-25
MATHEMATICS 146
Complete the following table:
Parallelogram Base Height Area Perimeter
(a) 5 units 3 units 15 sq units
(b)
(c)
(d)
(e)
(f)
(g)
Y ou will find that all these parallelograms have equal areas but different perimeters. Now ,
consider the following parallelograms with sides 7 cm and 5 cm (Fig 9.4).
Fig 9.4
Find the perimeter and area of each of these parallelograms. Analyse your results.
Y ou will find that these parallelograms have different areas but equal perimeters.
To find the area of a parallelogram, you need to know only the base and the
corresponding height of the parallelogram.
2024-25
PERIMETER AND AREA 147
TRY THESE
Find the area of following parallelograms:
(i) (ii)
(iii) In a parallelogram ABCD, AB = 7.2 cm and the perpendicular from C on AB is 4.5 cm.
9.2  AREA OF A TRIANGLE
A gardener wants to know the cost of covering the whole of a triangular
garden with grass.
In this case we need to know the area of the triangular region.
Let us find a method to get the area of a triangle.
Draw a scalene triangle on a piece of paper. Cut out the triangle.
Place this triangle on another piece of paper and cut out another
triangle of the same size.
So now you have two scalene triangles of the same size.
Are both the triangles congruent?
Superpose one triangle on the other so that they match.
Y ou may have to rotate one of the two triangles.
Now place both the triangles such that a pair of corresponding
sides is joined as shown in Fig 9.5.
Is the figure thus formed a parallelogram?
Compare the area of each triangle to the area of the
parallelogram.
Compare the base and height of the triangles with the base
and height of the parallelogram.
Y ou will find that the sum of the areas of both the triangles is
equal to the area of the parallelogram. The base and the height
of the triangle are the same as the base and the height of the
parallelogram, respectively .
Area of each triangle =
1
2
(Area of parallelogram)
=
1
2
(base × height) (Since area of a parallelogram = base × height)
=
1
2
( ) b h ×
 (or 
1
2
bh
, in short)
Fig 9.5
D
E
F
A
B
C
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MATHEMATICS 148
A
D C
4 cm
6 cm B
Fig 9.7
Fig 9.8
Fig 9.9
In the figure (Fig 9.6) all the triangles are on the base
AB = 6 cm.
What can you say about the height of each of the
triangles corresponding to the base AB?
Can we say all the triangles are equal in area? Y es.
Are the triangles congruent also? No.
We conclude that all the congruent triangles
are equal in area but the triangles equal in area
need not be congruent.
Consider the obtuse-angled triangle ABC of base 6 cm (Fig 9.7).
Its height AD which is perpendicular from the vertex A is outside the
triangle.
Can you find the area of the triangle?
EXAMPLE 1 One of the sides and the corresponding height of a
parallelogram are 4 cm and 3 cm respectively . Find the
area of the parallelogram (Fig 9.8).
SOLUTION Given that length of base (b) = 4 cm, height (h) = 3 cm
Area of the parallelogram = b × h
= 4 cm × 3 cm = 12 cm
2
EXAMPLE 2 Find the height ‘x’ if the area of the
parallelogram is 24 cm
2
 and the base is
4 cm.
SOLUTION  Area of parallelogram = b × h
Therefore,24 = 4 × x (Fig 9.9)
or
24
4
 = x or x = 6 cm
So, the height of the parallelogram is 6 cm.
TRY THESE
Fig 9.6
6 cm
1. Try the above activity with different types of triangles.
2. T ake different parallelograms. Divide each of the parallelograms into two triangles
by cutting along any of its diagonals. Are the triangles congruent?
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