Heron’s Formula

 

Area of triangle

 

Heron’s formula is used to find the area of a triangle ABC.

Let the sides of the triangle opposite to the vertices A, B, C be a, b, c.

 

Let the semi-perimeter be:

            

     Then the area of DABC

            

 

 

Heron of Alexandria

 

 

Proof

 

The proof of Heron’s formula is a good exercise of trigonometry (or geometry) and algebra.

Here is a proof:

 

The formulas we use are:

(1)        Area of triangle:

                            

(2)        Cosine Law:

                            

(3)        Square relation:

                             sin2 C + cos2 C = 1

    

From (3) and (2),

                    

                    

                    

From (1),

            

 

 

Verifications

 

You may wish to verify Heron’s formula for the following special cases:

 

(1)        A right-angled triangle with sides 3, 4, 5.

(2)        An equilateral triangle with sides a, a, a.