Derivation of Heron's / Hero's Formula for Area of Triangle

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derivation of formula
triangle
area of triangle
Heron

For a triangle of given three sides, say a, b, and c, the formula for the area is given by
 

$ A = \sqrt{s(s - a)(s - b)(s - c)} $

 

where s is the semi perimeter equal to P/2 = (a + b + c)/2.
 

Derivation of Heron's Formula
Triangle ABC of given three sidesArea of triangle ABC
$ A = \frac{1}{2}bh $       → equation (1)
 

From triangle ADB
$ x^2 + h^2 = c^2 $

$ x^2 = c^2 – h^2 $

$ x = \sqrt{c^2 - h^2} $
 

From triangle CDB
$ (b – x)^2 + h^2 = a^2 $

$ (b – x)^2 = a^2 – h^2 $

$ b^2 – 2bx + x^2 = a^2 – h^2 $
 

Substitute the values of x and x2
$ b^2 – 2b\sqrt{c^2 - h^2} + (c^2 – h^2) = a^2 – h^2 $

$ b^2 + c^2 – a^2 = 2b\sqrt{c^2 - h^2} $
 

Square both sides
$ (b^2 + c^2 – a^2)^2 = 4b^2(c^2 - h^2) $

$ \dfrac{(b^2 + c^2 – a^2)^2}{4b^2} = c^2 - h^2 $

$ h^2 = c^2 - \dfrac{(b^2 + c^2 – a^2)^2}{4b^2} $

$ h^2 = \dfrac{4b^2c^2 - (b^2 + c^2 – a^2)^2}{4b^2} $

$ h^2 = \dfrac{(2bc)^2 - (b^2 + c^2 – a^2)^2}{4b^2} $

$ h^2 = \dfrac{[ \, 2bc + (b^2 + c^2 – a^2) \, ][ \, 2bc - (b^2 + c^2 – a^2) \, ]}{4b^2} $

$ h^2 = \dfrac{[ \, 2bc + b^2 + c^2 – a^2 \, ][ \, 2bc - b^2 - c^2 + a^2 \, ]}{4b^2} $

$ h^2 = \dfrac{[ \, (b^2 + 2bc + c^2) – a^2 \, ][ \, a^2 - (b^2 - 2bc + c^2) \, ]}{4b^2} $

$ h^2 = \dfrac{[ \, (b + c)^2 – a^2 \, ] \cdot [ \, a^2 - (b - c)^2 \, ]}{4b^2} $

$ h^2 = \dfrac{[ \, (b + c) + a \, ][ \, (b + c) - a \, ] \cdot [ \, a + (b - c) \, ][ \, a - (b - c) \, ]}{4b^2} $

$ h^2 = \dfrac{(b + c + a)(b + c - a)(a + b - c)(a - b + c)}{4b^2} $

$ h^2 = \dfrac{(a + b + c)(b + c - a)(a + c - b)(a + b - c)}{4b^2} $

$ h^2 = \dfrac{(a + b + c)(a + b + c - 2a)(a + b + c - 2b)(a + b + c - 2c)}{4b^2} $

$ h^2 = \dfrac{P(P - 2a)(P - 2b)(P - 2c)}{4b^2} $   note: P = perimeter

$ h = \dfrac{\sqrt{P(P - 2a)(P - 2b)(P - 2c)}}{2b} $
 

Substitute h to equation (1)
$ A = \frac{1}{2}b\dfrac{\sqrt{P(P - 2a)(P - 2b)(P - 2c)}}{2b} $

$ A = \frac{1}{4}\sqrt{P(P - 2a)(P - 2b)(P - 2c)} $

$ A = \sqrt{\frac{1}{16}P(P - 2a)(P - 2b)(P - 2c)} $

$ A = \sqrt{\dfrac{P}{2} \left( \dfrac{P - 2a}{2} \right)\left( \dfrac{P - 2b}{2} \right)\left( \dfrac{P - 2c}{2} \right)} $

$ A = \sqrt{\dfrac{P}{2} \left( \dfrac{P}{2} - a \right)\left( \dfrac{P}{2} - b \right)\left( \dfrac{P}{2} - c \right)} $
 

Recall that P/2 = s. Thus,

$ A = \sqrt{s(s - a)(s - b)(s - c)} $

 

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