Area of a Triangle by Heron's Formula

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  • Collinearity of three points

Notes


Heron was born in about 10AD possibly in Alexandria in Egypt. He worked in applied mathematics. His geometrical works deal largely with problems on mensuration written in three books. In this book, Heron has derived the famous formula for the area of a triangle in terms of its three sides.
The formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as:

Area of a triangle = `sqrt (s(s-a) (s - b)(s-c))`

where a, b and c are the sides of the triangle, and s = semi-perimeter, i.e., half the perimeter of the triangle = `(a + b + c) /2`,
For instances, Let us take a = 40 m, b = 24 m, c = 32 m,
so that we have s = `(40 + 24 + 32 )/ 2` m
= 48 m 
s - a = (48 - 40) m = 8m
s - b = (48 - 24)m = 24 m 
s - c = (48 - 32 ) m = 16 m  
Therefore, area of the park ABC
`= sqrt (s(s - a) (s - b) (s - c))`
`= sqrt (48 * 8 * 24 * 16) m^2 = 384 m^2`

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