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# Herons Formula : Chapter Notes, Class 9 Mathematics Class 9 Notes | EduRev

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## Class 9 : Herons Formula : Chapter Notes, Class 9 Mathematics Class 9 Notes | EduRev

``` Page 1

HERONS FORMULA
Measuring the length, area and volume of plane figures is
called ‘Mensuration’.

Perimeter
Perimeter of a figure is the length of the boundary of that
figure.
A
B
C D

Perimeter = Total Length Of Boundary Of ABCD
= AB + BC + CD + DA

Area
Area of a figure is the total area enclosed by that figure.

Area = Area Enclosed By ABCD

Triangle

L B C

Perimeter = a + b + c
Where
a, b, c  Sides Of Triangle

Area
1. Area = Altitude Base
2
1
× ×

2. Area = c) (s b) (s a) (s s - - -
Where
a, b, c  Sides Of Triangle
s
2
c b a + +

Special Triangle Cases
There are few formulas, which can be applied only in case of a
particular type of triangle, such as right angled triangle,
isosceles triangle, equilateral triangle etc.

Note
These are not general formulas and can not be applied to
every type of triangle.
1. Right Angled Triangle

B C

Perimeter = a + b + c

Area =
2
1
× Base × Altitude
=
2
1
× a × b

Pythagoras Theorem
Hypotenuse
2
= Perpendicular
2
+ Base
2

2. Right Angled Isosceles Triangle

B C

Perimeter = 2a + b
Where
a  Equal Side
b   Unequal Side

Proof
Perimeter = Equal Side1 + Equal Side2 + Unequal Side
= (2 × Equal Side) + Unequal Side

Area =
2
a
2
1

Proof
Area =
2
1
× Base × Altitude
= a a
2
1
× ×
=
2
a
2
1

Hypotenuse = a 2

Proof
Hypotenuse
2
= Perpendicular
2
+ Base
2

Page 2

HERONS FORMULA
Measuring the length, area and volume of plane figures is
called ‘Mensuration’.

Perimeter
Perimeter of a figure is the length of the boundary of that
figure.
A
B
C D

Perimeter = Total Length Of Boundary Of ABCD
= AB + BC + CD + DA

Area
Area of a figure is the total area enclosed by that figure.

Area = Area Enclosed By ABCD

Triangle

L B C

Perimeter = a + b + c
Where
a, b, c  Sides Of Triangle

Area
1. Area = Altitude Base
2
1
× ×

2. Area = c) (s b) (s a) (s s - - -
Where
a, b, c  Sides Of Triangle
s
2
c b a + +

Special Triangle Cases
There are few formulas, which can be applied only in case of a
particular type of triangle, such as right angled triangle,
isosceles triangle, equilateral triangle etc.

Note
These are not general formulas and can not be applied to
every type of triangle.
1. Right Angled Triangle

B C

Perimeter = a + b + c

Area =
2
1
× Base × Altitude
=
2
1
× a × b

Pythagoras Theorem
Hypotenuse
2
= Perpendicular
2
+ Base
2

2. Right Angled Isosceles Triangle

B C

Perimeter = 2a + b
Where
a  Equal Side
b   Unequal Side

Proof
Perimeter = Equal Side1 + Equal Side2 + Unequal Side
= (2 × Equal Side) + Unequal Side

Area =
2
a
2
1

Proof
Area =
2
1
× Base × Altitude
= a a
2
1
× ×
=
2
a
2
1

Hypotenuse = a 2

Proof
Hypotenuse
2
= Perpendicular
2
+ Base
2

=
2 2
a a +       [Perpendicular = Base]
=
2 2
a a +
=
2
2a
= a 2

3. Equilateral Triangle
B C

Perimeter = 3a
Where
a  Equal Sides

Proof
Perimeter = Side1 + Side2 + Side3
= a + a + a
= 3a

Area =
2
a
4
3

Proof
Area = Altitude Base
2
1
× ×
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- × ×
2
2
2
a
a a
2
1

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- =
2
2
2
a
a Altitude Theorem, Pythagorus By
=
4
a
a a
2
1
2
2
- × ×
=
4
3a
a
2
1
2
× ×
=
2
3 a
a
2
1
× ×
=
2
a
4
3

Altitude = a
2
3

Proof
In right angled ? ABD,
Altitude =
2
2
2
a
a
?
?
?
?
?
?
?
?
-
=
4
a
a
2
2
-
=
4
3a
2

= a
2
3

A
B
C D

Perimeter  =  a + b + c + d
Where
a, b, c, d  Sides Of Trapezium

Area = ( ) ( ) ?B CD Area ? ABD Area +

= ( ) ( ) ( )
1 1 1 1 1 1 1
c s b s a s s - - -
+ ( ) ( ) ( )
2 2 2 2 2 2 2
c s b s a s s - - -

There are few formulas, which can be applied only in case of a
particular type of quadrilateral, such as rectangle, square,
parallelogram etc.

1. Trapezium

A B
C D

Perimeter =   a + b + c + d
Where
a, b, c, d  Sides Of Trapezium

Area =  Altitude ) S (S
2
1
2 1
× + ×
Where

1
S ,
2
S  Parallel Sides Of Trapezium

Proof
Area ( ) ( ) ( ) ?B CD Area ? ABD Area ABCD + =
=
?
?
?
?
?
?
?
?
+ × +
?
?
?
?
?
?
?
?
× × h S
2
1
h S
2
1
2 1

= ( )
2 1
S S h
2
1
+ × ×

2. Parallelogram
A B
C D

Perimeter = ( ) b a 2 +
Where

Page 3

HERONS FORMULA
Measuring the length, area and volume of plane figures is
called ‘Mensuration’.

Perimeter
Perimeter of a figure is the length of the boundary of that
figure.
A
B
C D

Perimeter = Total Length Of Boundary Of ABCD
= AB + BC + CD + DA

Area
Area of a figure is the total area enclosed by that figure.

Area = Area Enclosed By ABCD

Triangle

L B C

Perimeter = a + b + c
Where
a, b, c  Sides Of Triangle

Area
1. Area = Altitude Base
2
1
× ×

2. Area = c) (s b) (s a) (s s - - -
Where
a, b, c  Sides Of Triangle
s
2
c b a + +

Special Triangle Cases
There are few formulas, which can be applied only in case of a
particular type of triangle, such as right angled triangle,
isosceles triangle, equilateral triangle etc.

Note
These are not general formulas and can not be applied to
every type of triangle.
1. Right Angled Triangle

B C

Perimeter = a + b + c

Area =
2
1
× Base × Altitude
=
2
1
× a × b

Pythagoras Theorem
Hypotenuse
2
= Perpendicular
2
+ Base
2

2. Right Angled Isosceles Triangle

B C

Perimeter = 2a + b
Where
a  Equal Side
b   Unequal Side

Proof
Perimeter = Equal Side1 + Equal Side2 + Unequal Side
= (2 × Equal Side) + Unequal Side

Area =
2
a
2
1

Proof
Area =
2
1
× Base × Altitude
= a a
2
1
× ×
=
2
a
2
1

Hypotenuse = a 2

Proof
Hypotenuse
2
= Perpendicular
2
+ Base
2

=
2 2
a a +       [Perpendicular = Base]
=
2 2
a a +
=
2
2a
= a 2

3. Equilateral Triangle
B C

Perimeter = 3a
Where
a  Equal Sides

Proof
Perimeter = Side1 + Side2 + Side3
= a + a + a
= 3a

Area =
2
a
4
3

Proof
Area = Altitude Base
2
1
× ×
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- × ×
2
2
2
a
a a
2
1

?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
- =
2
2
2
a
a Altitude Theorem, Pythagorus By
=
4
a
a a
2
1
2
2
- × ×
=
4
3a
a
2
1
2
× ×
=
2
3 a
a
2
1
× ×
=
2
a
4
3

Altitude = a
2
3

Proof
In right angled ? ABD,
Altitude =
2
2
2
a
a
?
?
?
?
?
?
?
?
-
=
4
a
a
2
2
-
=
4
3a
2

= a
2
3

A
B
C D

Perimeter  =  a + b + c + d
Where
a, b, c, d  Sides Of Trapezium

Area = ( ) ( ) ?B CD Area ? ABD Area +

= ( ) ( ) ( )
1 1 1 1 1 1 1
c s b s a s s - - -
+ ( ) ( ) ( )
2 2 2 2 2 2 2
c s b s a s s - - -

There are few formulas, which can be applied only in case of a
particular type of quadrilateral, such as rectangle, square,
parallelogram etc.

1. Trapezium

A B
C D

Perimeter =   a + b + c + d
Where
a, b, c, d  Sides Of Trapezium

Area =  Altitude ) S (S
2
1
2 1
× + ×
Where

1
S ,
2
S  Parallel Sides Of Trapezium

Proof
Area ( ) ( ) ( ) ?B CD Area ? ABD Area ABCD + =
=
?
?
?
?
?
?
?
?
+ × +
?
?
?
?
?
?
?
?
× × h S
2
1
h S
2
1
2 1

= ( )
2 1
S S h
2
1
+ × ×

2. Parallelogram
A B
C D

Perimeter = ( ) b a 2 +
Where

a  Side1
b   Side2
Proof
Perimeter = a + b + a + b
= 2a + 2b
= ( ) b a 2 +

Area = Base × Altitude

3. Rhombus

A B
C D

Perimeter = 4a

Proof
Perimeter = a + a + a + a    [Sides of rhombus are equal]
= 4a

Area =
2 1
d d
2
1
× ×
Where
d1  Diagonal1
d2  Diagonal2

4. Rectangle

A B
C D

Perimeter = ( ) b a 2 +
Where
a  Side1
b   Side2

Proof
Perimeter = a + b + a + b
= 2a + 2b
= ( ) b a 2 +

Diagonal =
2 2
b a +      [By Pythagorus Theorm]

5. Square

A B
C D

Perimeter = 4a
Where
a  Side

Proof
Perimeter = a + a + a + a  [Sides of square are equal]
= 4a
Area =
2
a

Proof
= a a ×    [Sides of square are equal]
=
2
a

Diagonal = a 2

Proof
In right angled ? ABD,
d =
2 2
a a +     [Pythagoras Theorem]
=
2
2a
= 2 a

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