HERON'S FORMULA TO FIND AREA OF TRIANGLE

Scalene triangle is a triangle with all sides of different lengths.

All angles are different, too.

So no sides are equal and no angles are equal.

We can use Heron's formula to find area of a scalene triangle. 

Heron's Formula for Area of Scalene Triangle : 

=  √[s(s - a)(s - b)(s - c)]

where

S  =  (a + b + c) / 2

Here a, b and c are side lengths of the triangle. 

Practice Problems

Problem 1 :

Using Heron’s formula to find the area of a triangle whose side lengths are

10 cm, 24 cm and 26 cm

Solution :

a = 10 cm, b = 24 cm and c = 26 cm

s  = (a + b + c)/2

s  =  (10 + 24 + 26)/2

s  =  60/2  =  30

s - a  =  30 - 10  =  20

s - b  =  30 - 24  =  6

s - c  =  30 - 26  =  4

 √s(s − a)(s − b)(s − c)   =   √30(20)(6)(4) 

   =  120 cm²

Problem 2 :

Using Heron’s formula to find the area of a triangle whose side lengths are

1.8 m, 8 m, and 8.2 m

Solution : 

a = 1.8 m, b = 8 m and c = 8.2 m

s  = (a + b + c)/2

s  =  (1.8 + 8 + 8.2)/2

s  =  18/2  =  9

s - a  =  9 - 1.8  =  7.2

s - b  =  9 - 8  =  1

s - c  =  9 - 8.2  =  0.8

 √s(s − a)(s − b)(s − c)   =   √9(7.2)(1)(0.8) 

   =  7.2 m²

Problem 3 :

The sides of the triangular ground are 22 m, 120 m and 122 m. Find the area and cost of levelling the ground at the rate of ₹ 20 per m².

Solution :

a = 22 m, b = 120 m and c = 122 m

s  = (a + b + c)/2

s  =  (22 + 120 + 122)/2

s  =  264/2  =  132

s - a  =  132 - 22  = 110

s - b  =  132 - 120  =  12

s - c  =  132 - 122  =  10

 √s(s − a)(s − b)(s − c)   =   √132(110)(12)(10) 

=   √(11 ⋅ 12 ⋅ 11 ⋅ 10 ⋅ 12 ⋅ 10)

   =  11 ⋅ 12 ⋅ 10 

=  1320 m²

Cost of leveling the ground  =  ₹ 20 per m².

Required cost  =  20(1320)

 =  ₹ 26400

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