Heron’s Formula with Solved Examples

Heron’s formula was known after the hero of Alexandria who was a greek Engineer and Mathematician in 10 – 70 AD. In this formula, he gave the area of the triangle in terms of the length of its sides. The length of its sides is symbolised as a,b, and c. Semiperimeter of the triangle and side lengths are used to find the heron’s formula. Semi perimeter is denoted as “s”.

The formula for area of triangle is:

Area of triangle = √(s(s-a)(s-b)(s-c)

Semi-perimeter, s = (a + b + c)/2

Area of the Triangle using Heron’s Formula?

The steps to find the area by Heron’s formula are:

• Step 1: First find the perimeter of the triangle.
• Step 2: Then find the semi-perimeter of that triangle by halving it.
• Step 3: Then find the area of the triangle by Heron’s formula which is √(s(s – a)(s – b)(s – c)).
• Step 4: Once the value is found out, write down the unit at the end (For example, m2, cm2, or in2)

Heron’s Formula for Equilateral Triangle

Area of triangle = √(s(s-a)(s-b)(s-c)

Since, a = b = c

So,

A = √[s(s-a)³]

Heron’s Formula for Isosceles Triangle

Area of triangle = √[s(s – a)(s – b)(s – c)]

Now, substituting the sides of an isosceles triangle,

Area of triangle = √[s(s – a)(s – a)(s – b)]

A = √[s(s – a)²(s – b)⦐

Applications of Heron’s Formula

The following two are the applications of Heron’s formula :

• It is applicable when there is different types of triangle 
• Also, it is applicable to determine the area of quadrilateral

Examples

Example 1: A triangle PQR has sides 5 cm, 14 cm and 17 cm. Find the area of the triangle.

Solution:

Semiperimeter of triangle PQR, s = (5+14+17)/2 = 36/2 = 18

By using heron’s formula, 

A = √[s(s-a)(s-b)(s-c)]

Hence, A = √[18(18-5)(18-14)(18-17)] = √(18 x 13 x 4 x 1) =  √936 = 30.6sq.cm

Example 2: Find the area of the triangle whose sides measure 12 cm, 14 cm and 20 cm.  Also, determine the length of the altitude on the side which measures 14 cm.

Solution: s = (a + b + c)/2

                   =23

Area of Triangle= √[s(s-a)(s-b)(s-c)]

 A = √[23(23-12)(23-14)(23-20)]

√(23 x 11 x 9 x 3)

=  82.65square cm

Taking 14 cm as the base length we need to find the height

Area, A = 1/2 x base x height

1/2  x 14 x h = 82.65   or h = 165.3/14 = 11.80 cm  

Thus the area of the triangle using heron’s formula is 82.65 sq. cm.