Area of a triangle (Heron’s Formula )

Heron’s formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in (CE - ). This formula is used only when all the three sides are known. Theorem . : In △ ABC , △ = p s ( s − a )( s − b )( s − c ) where s is the semi-perimeter of △ ABC .

Proof. △ = ab sin C = ab sin C cos C = ab r ( s − a )( s − b ) ab r s ( s − c ) ab p s ( s − a )( s − b )( s − c ) Trigonometry (i) Using Heron’s formula , Pythagorean theorem can be proved for right triangle and conversely, using Pythagorean theorem for right triangle one can establish Heron’s area formula. (ii) If area of a triangle is given as an integer, then Heron’s formula is useful in finding triangles with integer sides. (iii) If the perimeter of a triangle is fixed, then Heron’s formula is useful for finding triangles having integer area and integer sides.

For example, if the perimeter of a triangle is m, then there is a triangle with sides m, m, m and area m . Example . In a △ ABC , prove that b sin C + c sin B = bc sin A . The Sine formula is, a sin A = b sin B = c