Areas (heron's Formula)

       
           

For calculation of area of triangle heron's formula can be used but this method is suitable when lengths of all three sides are given.

Let x, y, z the length of three sides of triangle then the formula is given by;

 Area = √ [s (s – x) (s – y) (s – z)];where s = (x + y + z)/2.

This formula was given by Heron of Alexandria and proof of this formula was found in his book ‘Metrica’ which was written in A.D. 60. Heron was among the greatest mathematicians of antiquity he also worked on quadrilaterals and higher-order polygons. Although the herons formula is deceptively simple, heron's utilized the properties of inscribed quadrilaterals and right triangle in the original proof.

Now let's see the modern proof of Heron s formula which is not similar to the earlier proof of heron's formula which he gave in his book ‘metrica’ let x, y, z be the sides of the triangle and A, B, C be angles opposite to those sides. Then we will have

               x2 + y2 – z2,

cos Ĉ = -----------------

                  2xy,

Now by the law of cosines, we will get the algebraic statement:

                                     √[4 xy- ( x+ y2 – z2)2],

sin Ĉ = √(1-cosĈ = ------------------------------

                                                      2xy,

The altitude of triangle on the base has length y∙sin (C) and then it follows:

area = ½ (base) (altitude),

= 1/2 xy sin Ĉ

= 1/4 √[4 xy- (x+ y2 – z2)2],

= ¼ √[2xy - (x+ y2 – z2)][2xy + ( x+ y2 – z2)],

= ¼ √ [z2- (x - y)2][(x + y)2 – z2],

 = ¼ √ [z- (x - y)][z + (x - y)][(x + y) - z][(x + y) + z],

= √ [z- ( x - y)][z + ( x - y)][( x+ y) - z][( x + y) + z] / (1/16),

 = √[z- ( x - y)]   √[z + ( x - y)]           √[( x+ y) - z]                     √[( x + y) + z]

     ---------           ------------                 -----------                      ----------------

          √2               √2                             √2                                       √2

 

 

    √(z + y - x)  √(y+z-x)      √(x+z-y)              √(x+y-z)

=      -------        ----              ---------             ----------   

         √2           √2                √2                       √2

 

= √ [s (s – x) (s – y) (s – z)].

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