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Title: History/By Topic/Science/Engineering and Technology/Biography/Heron of Alexandria - Mathworld - Heron's Formula An important theorem in plane geometry. Describes it in detail while relating other formulas.
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Heron's Formula -- from Wolfram MathWorld Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book 12,273 entries Last updated: Mon Oct 6 2008 Created, developed, and nurtured by Eric Weisstein at Wolfram Research Geometry > Plane Geometry > Triangles > Triangle Properties >Interactive Entries > Interactive Demonstrations >Heron's FormulaDOWNLOAD Mathematica NotebookAn important theorem in plane geometry, also known as Hero's formula. Given the lengths of the sides a, b, and c and the semiperimeter s=1/2(a+b+c) (1)of a triangle, Heron's formula gives the area Delta of the triangle as Delta=sqrt(s(s-a)(s-b)(s-c)). (2)Heron's formula may be stated beautifully using a Cayley-Menger determinant as -16Delta^2=|0 a b c; a 0 c b; b c 0 a; c b a 0|=|0 1 1 1; 1 0 c^2 b^2; 1 c^2 0 a^2; 1 b^2 a^2 0|. (3)Another highly symmetrical form is given by (4Delta)^2=[a^2 b^2 c^2][-1 1 1; 1 -1 1; 1 1 -1][a^2; b^2; c^2] (4)(Buchholz 1992).SoddyCirclesExpressing the side lengths a, b, and c in terms of the radii a^', b^', and c' of the mutually tangent circles centered on the triangle vertices (which define the Soddy circles),a=b^'+c^'(5)b=a^'+c^'(6)c=a^'+b^',(7)gives the particularly pretty form Delta=sqrt(a^'b^'c^'(a^'+b^'+c^')). (8)Heron's proof (Dunham 1990) is ingenious but extremely convoluted, bringing together a sequence of apparently unrelated geometric identities and relying on the properties of cyclic quadrilaterals and right triangles. Heron's proof can be found in Proposition 1.8 of his work Metrica (ca. 100 BC-100 AD). This manuscript had been lost for centuries until a fragment was discovered in 1894 and a complete copy in 1896 (Dunham 1990, p. 118). More recently, writings of the Arab scholar Abu'l Raihan Muhammed al-Biruni have credited the formula to Heron's predecessor Archimedes prior to 212 BC (van der Waerden 1961, pp. 228 and 277; Coxeter and Greitzer 1967, p. 59; Kline 1990; Bell 1986, p. 58; Dunham 1990, p. 127).A much more accessible algebraic proof proceeds from the law of cosines, cosA=(b^2+c^2-a^2)/(2bc). (9)Then sinA=(sqrt(-a^4-b^4-c^4+2b^2c^2+2c^2a^2+2a^2b^2))/(2bc), (10)givingDelta=sqrt(s(s-a)(s-b)(s-c))(11)=1/4sqrt((2ab)^2-(a^2+b^2-c^2)^2)(12)=1/2bcsinA(13)=1/4sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c))(14)(15)=1/4sqrt(2(b^2c^2+c^2a^2+a^2b^2)-(a^4+b^4+c^4))(16)(Coxeter 1969). Heron's formula contains the Pythagoreantheorem as a degenerate case.SEE ALSO: Brahmagupta's Formula, Bretschneider's Formula, Cayley-Menger Determinant, Heronian Tetrahedron, Heronian Triangle, Soddy Circles, SSS Theorem, TriangleREFERENCES:Bell, E. T. Men of Mathematics. New York: Simon and Schuster, p. 58, 1986.Buchholz, R. H. "Perfect Pyramids." Bull. Austral. Math. Soc. 45, 353-368, 1992. http://www.geocities.com/teufel_pi/papers/perfectpyramids.pdf.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 12, 1969.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 59, 1967.Dunham, W. "Heron's Formula for Triangular Area." Ch. 5 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 113-132, 1990.Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford, England: Oxford University Press, 1990.MathPages. "Heron's Formula and Brahmagupta's Generalization." http://www.mathpages.com/home/kmath196.htm.Pappas, T. "Heron's Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 62, 1989.van der Waerden, B. L. Science Awakening. Oxford, England: Oxford University Press, pp. 228 and 277, 1961.CITE THIS AS:Weisstein, Eric W. "Heron's Formula."From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HeronsFormula.htmlSendContact the MathWorld Team© 1999-2008 Wolfram Research, Inc. | Terms of UseHeron's Formula in the New! Interactive mathematics--The Wolfram Demonstrations ProjectTriangles in the Coordinate PlaneTriangles in the Coordinate PlaneTriangles in the Coordinate PlaneLinked SlidersLinked SlidersLinked SlidersSolving Oblique TrianglesSolving Oblique TrianglesSolving Oblique TrianglesWear Your Math Proudly!Get your own MathWorld T-shirt »  Other Wolfram Web Resources » Wolfram Research Mathematica Home Page Mathematica Documentation Center Wolfram Demonstrations Project Online Integrator Functions Site Wolfram Tones Wolfram Science Wolfram Blog more…
 

An

important

theorem

in

plane

geometry.

Describes

it

in

detail

while

relating

other

formulas.

http://mathworld.wolfram.com/HeronsFormula.html

Mathworld - Heron's Formula 2008 October

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An important theorem in plane geometry. Describes it in detail while relating other formulas.

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