Heron's formula
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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
- <math>\mathrm{area} = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\,<math>
where s is the triangle's semiperimeter:
- <math>s=\frac{a+b+c}{2}<math>
(see also square root). Heron's formula can also be written
- <math>\mathrm{area}={\ \sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)\,}\ \over 4}.\,<math>
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Numerical stability
Heron's formula as given above is numerically unstable for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that: a ≥ b ≥ c and computing
- <math>S = 1/4\sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}<math>
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 Hindus.
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
- <math>\cos(C) = \frac{a^2+b^2-c^2}{2ab}<math>
by the law of cosines. From this we get with some algebra
- <math>\sin(C) = \sqrt{1-\cos^2(C)} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}<math>.
The altitude of the triangle on base a has length bsin(C), and it follows
- <math>S = \frac{1}{2} (\mbox{base}) (\mbox{altitude})<math>
- <math>\qquad = \frac{1}{2} ab\sin(C)<math>
- <math>\qquad = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2}<math>
- <math>\qquad = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.<math>
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,
- <math> 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} } <math> illustrates its similarity to Tartaglia's formula for the volume of a four-simplex.
See also
External links
- MathWorld entry on Heron's Formula (http://mathworld.wolfram.com/HeronsFormula.html)
- Semiperimeter, incircle and excircles of a triangle (http://agutie.homestead.com/files/semiperimeterexcirclesinc1.htm) by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".
- A Proof of the Pythagorean Theorem From Heron's Formula (http://www.cut-the-knot.org/pythagoras/herons.shtml)de:Satz des Heron
Heron's formula fr:Formule de Héron it:Formula di Erone ja:ヘロンの公式 nl:Formule van Heron pl:Wzór Herona ru:Формула Герона fi:Heronin kaava zh:海伦公式