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

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

where s is the triangle's semiperimeter:

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

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

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

History

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The formula is credited to Heron of Alexandria in the 1st century, 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 was known long before.

Proof

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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

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

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

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

The altitude of the triangle on base a has length bsin(C), and it follows

{| $S\backslash ,$ $=\; \backslash frac\{1\}\{2\}\; (\backslash mbox\{base\})\; (\backslash mbox\{altitude\})$ $=\; \backslash frac\{1\}\{2\}\; ab\backslash sin(C)$ $=\; \backslash frac\{1\}\{4\}\backslash sqrt\{4a^2\; b^2\; -(a^2\; +b^2\; -c^2)^2\}$ $=\backslash sqrt\{s\backslash left(s-a\backslash right)\backslash left(s-b\backslash right)\backslash left(s-c\backslash right)\}.$

Here the algebra in the last step was omitted.

Numerical stability

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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

$S\; =\; \backslash frac\{1\}\{4\}\backslash sqrt\{(a+(b+c))\; (c-(a-b))\; (c+(a-b))\; (a+(b-c))\}$

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Generalizations

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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,

$S\; =\; \backslash frac\{1\}\{4\}\; \backslash sqrt\{\; \backslash begin\{vmatrix\}$

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 four-simplex.

See also

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• Synthetic geometry
• Heronian triangle

References

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External links

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• MathWorld entry on Heron's Formula
• Semiperimeter, incircle and excircles of a triangle by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".
• A Proof of the Pythagorean Theorem From Heron's Formula at cut-the-knot
• Interactive applet and area calculator using Heron's Formula

Euclidean plane geometry | Mathematical theorems | Proofs | Triangles

معادلة هيرون | Herons formel | Satz des Heron | Τύπος του Ήρωνα | Fórmula de Herón | Formule de Héron | 헤론의 공식 | Formula di Erone | נוסחת הרון | Formule van Heron | ヘロンの公式 | Wzór Herona | Teorema de Heron | Формула Герона | Heronin kaava | Herons formel | 海伦公式