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.