Introduction to Heron Area Formula

Heron area formula is the formula used to find the area of triangle. The heron area formula is created by Heron of Alexandria. The proof of heron area formula can be proved by using Pythagoras theorem found in his book, Metrica. Heron area formula is also known as Hero's formula. The sides of triangle are a,b and c.

The semi perimeter is given as,

s= (a+b+c)/2 or perimeter/2.

The heron area formula is given as,

Area= (s(s-a) (s-b) (s-c).
Examples Involving Herons Area Formula

Example1:

What is the area of a triangle where every side is 5 long?

Given:

a = b = c = 5

Solution:

Semi perimeter is given as,

s= (a+b+c)/2 or perimeter/2.

s= (5+5+5)/2

s= (15)/2

s=7.5

The heron area formula is given as,

Area= (s(s-a) (s-b) (s-c)),

Area= (7.5(7.5-5) (7.5-5) (7.5-5))

Area= (7.5(2.5) (2.5) (2.5))

Area = (117.1875)

Area = 10.825

Example2:

What is the area of a triangle where a=5, b=7, c=9?

Given:

a = 5, b = 7, c = 9

Solution:

Semi perimeter is given as,

s= (a+b+c)/2 or perimeter/2.

s= (5+7+9)/2

s= (21)/2

s=10.5

The heron area formula is given as,

Area= (s(s-a) (s-b) (s-c)),

Area= (10.5(10.5-5) (10.5-7) (10.5-9))

Area= (7.5(5.5) (3.5) (1.5))

Area = (216.5625)

Area = 14.71
Herons Area Fromula Example 3:

What is the area of a triangle where a=3, b=6 and s=9?

Given:

a = 3, b = 6, s = 9

Solution:

Semi perimeter is given as,

s= (a+b+c)/2 or perimeter/2.

9= (3+6+c)/2

9*2 = (9+c)

18 = 9 +c

C=9

The heron area formula is given as,

Area= (s(s-a) (s-b) (s-c)),

Area= (9(9-5) (9-7) (9-9))

Area= (9(4 (2) (0))

Area = (0)

Area = 0

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Example4:

What is the area of a triangle where a=5, c=9 and s=10?

Given:

a=5, c=9 and s=10

Solution:

Semi perimeter is given as,

s= (a+b+c)/2 or perimeter/2.

10= (5+b+9)/2

10*2 = (14+b)

20 = 14 +c

C=6

The heron area formula is given as,

Area= (s(s-a) (s-b) (s-c)),

Area= (10(10-5) (10-6) (10-9))

Area= (10(5 (4) (1))

Area = (200)

Area = 14.14