Heron’s formula is named after Hero of Alexendria. Heron’s formula is a formula that can be used to find the area of a triangle, when given its three side lengths.

Heron’s formula

The area of a triangle whose side lengths are a,b and c is given by

A = √s(s−a)(s−b)(s−c)

where s semi-perimeter of the triangle.
s = (perimeter of the triangle)2
s = (a + b + c)2

Problems

Example1: Use Heron’s formula to find the area of triangle ABC, if AB=3,BC=2,CA=4
Solution:
Step 1: Calculate the semi perimeter, S
s = (3 + 2 + 4)2 = 4.5

Step 2: Area of triangle

A = √s(s−a)(s−b)(s−c)

A = √4.5(4.5−3)(4.5−2)(4.5−4)
A = √8.437
A = 2.9

Example2:Given △ABC, with an area of 8.94 square units, a perimeter of 16 units and side lengths AB=3 and CA=7, what is BC ?
Solution:
Given s = 16 units , A = 8.94 square units
s = perimeter2 = 162
s = 8
Using Heron’s formula,
A = √S(S−AB)(S−BC)(S−CA)
8.94 = √8(8−3)(8−BC)(8−7)
8.94 = √8(5)(8−BC)(1)
square both sides
8.94² = 8(5)(8−BC)(1)
79.9236 = 40(8−BC)
BC = 6

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