NationMaster - Encyclopedia: Heron's formula

In geometry, Heron's (or Hero's) formula states that the area (A) of a triangle whose sides have lengths a, b, and c is Image File history File links Triangle_with_notations_2. ... Image File history File links Triangle_with_notations_2. ... For other uses, see Geometry (disambiguation). ... This article is about the physical quantity. ... For other uses, see Triangle (disambiguation). ...

where s is the semiperimeter of the triangle: The semiperimeter of a mathematical shape is defined as half of the shapes perimeter. ...

$s=frac{a+b+c}{2}.$

Heron's formula can also be written as:

$A={ sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b),} over 4}$

$A={ sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4),} over 4}$

$A={ sqrt{(a^2 + b^2 + c^2)^2 - 2(a^4 + b^4 + c^4),} over 4}.$

1 History
2 Proof
3 Proof using the Pythagorean theorem
4 Numerical stability
5 Generalizations
6 See also
7 References
8 External links

History

The formula is credited to Heron of Alexandria, and a proof can be found in his book, Metrica, written c. A.D. 60. It has been suggested that Archimedes knew the formula, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that it predates the reference given in the work. ^[1] Heros aeolipile Hero (or Heron) of Alexandria (c. ... For other uses, see Archimedes (disambiguation). ...

A formula equivalent to Heron's namely:

$A=frac1{2}sqrt{a^2 c^2 - left( frac{a^2+c^2-b^2}{2} right)^2}$

was discovered by the Chinese independently of the Greeks. It was published in Shushu Jiuzhang (“Mathematical Treatise in Nine Sections”), written by Qin Jiushao and published in A.D. 1247. The Mathematical Treatise in Nine Sections (Traditional Chinese: ; Simplified Chinese: ; Hanyu Pinyin: ShÃ¹shÅ« JiÇ”zhÄng; Wade-Giles: Shshu Chiuchang) is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. ... Châ€™in Chiu-Shao (ç§¦ä¹éŸ¶ or ç§¦ä¹åŠ, transcribed Qin Jiushao in pinyin) (ca. ...

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 This article is about the branch of mathematics. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... This article is about angles in geometry. ...

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

by the law of cosines. From this we get the algebraic statement: Fig. ...

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

The altitude of the triangle on base a has length bsin(C), and it follows In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ...

$A,$	$= frac{1}{2} (mbox{base}) (mbox{altitude})$
	$= frac{1}{2} absin(C)$
	$= frac{1}{4}sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2}$
	$= frac{1}{4}sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))}$
	$= frac{1}{4}sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)}$
	$= frac{1}{4}sqrt{(c -(a -b))((c +(a -b))((a +b) -c))((a +b) +c)}$
	$= sqrt{sleft(s-aright)left(s-bright)left(s-cright)}.$

The difference of two squares factorization was used in two different steps. In mathematics, the difference of two squares refers to the identity a2 âˆ’ b2 = (a + b)(a âˆ’ b) from elementary algebra. ...

Proof using the Pythagorean theorem

Triangle with altitude h cutting base c into d+(c−d).

Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as above, or to the incenter and one excircle of the triangle[1]. The following argument reduces Heron's formula directly to the Pythagorean theorem using only elementary means. In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

In the form $4 A 2 = 4 s (s - a)(s - b)(s - c)$ , Heron's formula reduces on the left to $(c h) 2$ , or

(c b) 2 - (c d) 2

using $b 2 - d 2 = h 2$ by the Pythagorean theorem, and on the right to In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

(s (s - a) + (s - b)(s - c)) 2

−

((s (s - a) - (s - b)(s - c)) 2

via the principle $(p + q) 2 - (p - q) 2 = 4 p q$ . It therefore suffices to show

c b = s (s - a) + (s - b)(s - c)

, and

c d = s (s - a) - (s - b)(s - c)

The former follows immediately by substituting $(a + b + c) / 2$ for $s$ and simplifying. Doing this for the latter reduces $s (s - a) - (s - b)(s - c)$ only as far as $(b 2 + c 2 - a 2) / 2$ . But if we replace $b 2$ by $d 2 + h 2$ and $a 2$ by $(c - d) 2 + h 2$ , both by Pythagoras, simplification then produces $c d$ as required.

Numerical stability

Heron's formula as given above is numerically unstable for triangles with a very small angle. A stable alternative^[2] involves arranging the lengths of the sides so that: a ≥ b ≥ c and computing In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...

$A = frac{1}{4}sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}.$

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

Generalizations

Heron's formula is 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. In both cases Heron's formula is obtained by setting one of the sides of the quadrilateral to zero. In geometry, Brahmaguptas formula formula finds the area of any quadrilateral. ... In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ... Given a general quadrilateral with sides of lengths , , , and , the area is given by (1) (2) (Coolidge 1939; Ivanov 1960; Beyer 1987, p. ... This article is about the geometric shape. ...

Heron's formula is also a special case of the formula of the area of the trapezoid based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero. This article is about the geometric figure. ...

Expressing Heron's formula with a determinant in terms of the squares of the distances between the three given vertices, In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... For distance between people, see proxemics. ...

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

illustrates its similarity to Tartaglia's formula for the volume of a three-simplex. Niccolo Fontana Tartaglia. ... For other uses, see Volume (disambiguation). ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ...

Another generalization of Heron's formula to polygons inscribed in a circle was discovered by David P. Robbins.^{[citation needed]}

References

^ Heron's Formula - from Wolfram MathWorld
^ http://http.cs.berkeley.edu/~wkahan/Triangle.pdf

Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press, 321-323.

External links

MathWorld entry on Heron's Formula
A Proof of the Pythagorean Theorem From Heron's Formula at cut-the-knot
Interactive applet and area calculator using Heron's Formula
Implementations of Heron's formula in various programming languages
J.H. Conway discussion on Heron's Formula
Kevin Brown's simplification of Heron's Pythagorean argument

Categories: Triangle geometry | Mathematical theorems | Articles containing proofs | Area

Hidden categories: All articles with unsourced statements | Articles with unsourced statements since November 2007

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

Results from FactBites:

Heron's Formula (145 words)
	You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been know for nearly 2000 years.
	The formula is credited to Hero (or Heron) of Alexandria, who was a Greek Engineer and Mathematician in 10 – 70 AD.
	In the calculator above I have also used a special formula to calculate the angles (to provide a complete solution).

More results at FactBites »

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