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Heron's Formula


An 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).

SoddyCircles

Expressing 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)

giving

Delta=1/2bcsinA
(11)
=1/4sqrt(2(b^2c^2+c^2a^2+a^2b^2)-(a^4+b^4+c^4))
(12)
=1/4sqrt((2ab)^2-(a^2+b^2-c^2)^2)
(13)
=1/4sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c))
(14)
=sqrt(s(s-a)(s-b)(s-c))
(15)

(Coxeter 1969). Heron's formula contains the Pythagorean theorem as a degenerate case.


See also

Brahmagupta's Formula, Bretschneider's Formula, Cayley-Menger Determinant, Heronian Tetrahedron, Heronian Triangle, Soddy Circles, SSS Theorem, Triangle

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References

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

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