## Thursday, August 19, 2010

### Heron's Formula

Pythagoras gets all the publicity but Heron was no slouch either... Heron was another ancient who is credited with devising a formula for computing the area of a triangle from knowing only the lengths of the 3 sides involved. The formula appears in a few different forms, 2 common ones below (a, b, and c represent the 3 sides of a triangle, and "s" is the perimeter value for same).
$A=\frac{1}{4}\sqrt{(a^2 + b^2 + c^2)^2 - 2(a^4 + b^4 + c^4)}.$
$A = \sqrt{s(s-a)(s-b)(s-c)}$

more on Heron's formula here:

http://en.wikipedia.org/wiki/Heron%27s_formula

An offshoot of Heron's formula is Brahmagupta's formula for the area of any 'cyclic'
quadrilateral (one that fits inside a circle):
$\sqrt{(s-a)(s-b)(s-c)(s-d)}$
or
     _________________________________
1/4 √(a+b+c-d) (a+b-c+d) (a-b+c+d) (-a+b+c+d)

#### 1 comment:

Pat B said...

"a, b, and c represent the 3 sides of a triangle, and "s" is the perimeter value for same)" I think the word perimeter should be semi-perimeter...