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
 
 
 
In quadrilateral ABCD, 
Perimeter = Total Length Of Boundary Of ABCD 
  = AB + BC + CD + DA 
 
Area 
Area of a figure is the total area enclosed by that figure. 
  
In quadrilateral ABCD, 
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
 
 
 
In quadrilateral ABCD, 
Perimeter = Total Length Of Boundary Of ABCD 
  = AB + BC + CD + DA 
 
Area 
Area of a figure is the total area enclosed by that figure. 
  
In quadrilateral ABCD, 
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
 
Quadrilateral 
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 - - - 
 
Special Quadrilateral Cases 
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
 
 
 
In quadrilateral ABCD, 
Perimeter = Total Length Of Boundary Of ABCD 
  = AB + BC + CD + DA 
 
Area 
Area of a figure is the total area enclosed by that figure. 
  
In quadrilateral ABCD, 
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
 
Quadrilateral 
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 - - - 
 
Special Quadrilateral Cases 
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 + 
 
Area = Length × Breadth 
 
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 
 Area = Breadth Lenglth × 
    = 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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