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.