Heron’s formula

Heron’s formula

  • In geometry, Heron’s formula, named after Hero of Alexandria gives us the area of any triangle whose three sides lengths are already known.
  • So, we can find area of triangle from their three sides also.
  • If a, b, c are the lengths of the three sides of the triangle then area of that triangle is given by,

Area of triangle = √s(s-a)(s-b)(s-c)

Where, s = (a + b + c)/ 2

  • This is the formula which is known as Heron’s Formula.

Derivation for Heron’s Formula:

  • Let us consider the ΔABC with three sides length a, b and c as shown in following figure.
  • We drawn the perpendicular AP on base BC and l(BC) = b, l(AP) = h.
  • Let x be the length of BP, l(BP) = x and hence l(PC) = (b – x).
  • In general, we know that area of triangle ABC having base b and height h is given by,

Area of triangle = ½ * base * height

 

Area of triangle = ½*b*h

In ΔABP, by Pythagoras theorem

Thus, a2 = x2 + h2

Hence, h2 = a2 – x2

 

In ΔAPC, by Pythagoras theorem

Thus, c2 = h2 + (b – x)2

Hence, h2 = c2 – (b2 -2bx + x2) = c2 – b2 + 2bx – x2

 

We equate the value of h2 from both triangles,

a2 – x2= c2 – b2 + 2bx – x2

a2 = c2 – b2 + 2bx

a2 – c2 + b2 = 2bx

thus, x = (a2 – c2 + b2)/ 2b

 

By putting the value of x in equation h2 = a2 – x2,

We get, h2 = a2 – [ (a2 – c2 + b2)/ 2b]2

Thus, h2 = [4a2b2 – (a2 – c2 + b2)2]/ 4b2

Hence, h2 = 1/4b2*[(2ab)2 – (a2 – c2 + b2)2]

 

By using the formula here, (A2 – B2) = (A – B) *(A + B)

We get, h2 = 1/4b2*[(2ab + a2 – c2 + b2) (2ab – a2 + c2 – b2)

Thus, h2 = 1/4b2*[ (a + b)2 – c2] [c2 – (a – b)2]

Hence, h2 = 1/4b2*[(a + b +c) (a + b – c)] [(c + a – b) (c – a + b)]

 

But, s = (a + b + c)/ 2

Thus, h2 = 1/4b2*[2s*(2s – 2c) *(2s – 2b) *(2s – 2a)]

Hence, h2 = 1/4b2*4[4*(s – a) *(s – b) *(s – c)]

h2 = 1/b2*[4*(s – a) *(s – b) *(s – c)]

h = 2/b*√[(s – a) *(s – b) *(s – c)]

 

Now, we know that area of triangle ABC = ½*base*height

= ½*b*h

=1/2*b*2/b*√ [(s – a) *(s – b) *(s – c)]

= √ [(s – a) *(s – b) *(s – c)]

 

Thus, area of triangle if the lengths of three sides of triangle are known is given by,

Area of triangle = √ [(s – a) *(s – b) *(s – c)]

This formula is known as Heron’s Formula.

Hence proved.


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