Area of Triangle ( Heron’s Formula )

 

Motivation:

                         Is it possible to find the area of a triangle by using the lengths of its three sides?  With three sides, we are able to find out the cosine value of each angle.  Via the value of cosine,  it is possible to find the value of sine for the given angle.  As a whole, it seems a solvable question.  Actually, the result is known as Heron’s Formula.

 

 

 

 

 

 

 

Theorem ( Heron’s Formula )

                    The lengths of three sides of    ABC are known as a, b, c .

           If    s= ½ (a+b+c),    then

 

                        ABC = 

 

        Proof:

 

                         From the relationship of trigonometric functions, we always have

 

                                             sin²C + cos²C = 1

 

                         And inside  ABC,  the value of sine should always be positive.

                                

                                      sinC =

 

 

                         Thus,

 

                          ABC = ½ ab sinC

                                     = ½ ab

 

                         And from cosine law, we have

                                               

                                              cosC =

 

                        ABC = ½ ab

                                        = ¼

                                        = ¼

                                        = ¼

                                        = ¼

                                        =