Heron's Formula

#REDIRECT Heron's formula

Heron's formula

In geometry, '''Heron's formula (also called Hero's formula''') states that the area of a triangle whose sides have lengths ''a'', ''b'' and ''c'' is :

\mathrm{area} = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\,

where ''s'' is the triangle's semiperimeter: :

s=\frac{a+b+c}{2}

(see also square root). Heron's formula can also be written :

\mathrm{area}={\ \sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)\,}\ \over 4}.\,

== Numerical stability == Heron's formula as given above is Numerical_stability for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that: ''a'' ≥ ''b'' ≥ ''c'' and computing :

S = 1/4\sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}

The brackets in the above formula are required in order to prevent numerical instability in the evaluation. == History == The formula is credited to Heron of Alexandria in the 1st century AD, and a proof can be found in his book ''Metrica''. It is now believed that Archimedes already knew the formula, and it is of course possible that it has been known long before. After the re-discovery of the Vedic Mathematics, it is now believed that this formula should be credited to the ancient India. == Proof == A modern proof, which uses algebra and trigonometry and is quite unlike the one provided by Heron, follows. Let ''a'', ''b'', ''c'' be the sides of the triangle and ''A'', ''B'', ''C'' the angles opposite those sides. We have :

\cos(C) = \frac{a^2+b^2-c^2}{2ab}

by the law of cosines. From this we get with some algebra :

\sin(C) = \sqrt{1-\cos^2(C)} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}

. The altitude (triangle) of the triangle on base ''a'' has length ''b''sin(C), and it follows :

S = \frac{1}{2} (\mbox{base}) (\mbox{altitude})

\qquad = \frac{1}{2} ab\sin(C)

\qquad = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2}

\qquad = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.

Here the simple algebra in the last step was omitted. == Generalizations == The formula is in fact a special case of Brahmagupta's formula for the area of a cyclic quadrilateral; both of which are special cases of Bretschneider's formula for the area of a quadrilateral. Expressing Heron's formula with a determinant in terms of the squares of the distances between the three given vertices, :

S =  \frac{1}{4} \sqrt{ \begin{vmatrix} 
  0 & a^2 & b^2 & 1 \\
a^2 & 0   & c^2 & 1 \\
b^2 & c^2 & 0   & 1 \\
  1 &   1 &   1 & 0
\end{vmatrix} }

illustrates its similarity to Tartaglia's formula for the volume of a Simplex. ==See also== * Synthetic geometry == External links == * [http://mathworld.wolfram.com/HeronsFormula.html MathWorld entry on Heron's Formula] *[http://agutie.homestead.com/files/semiperimeterexcirclesinc1.htm Semiperimeter, incircle and excircles of a triangle] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas". *[http://www.cut-the-knot.org/pythagoras/herons.shtml A Proof of the Pythagorean Theorem From Heron's Formula] Euclidean geometry Theorems Proofs Triangles Heron's formula

Heron's formula

how does making a>b>c change the formula? i can't see how it influences it.

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