Heron's formula
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In geometry, Heron's (or Hero's) formula, named after Heron of Alexandria,[1] states that the area T of a triangle whose sides have lengths a, b, and c is
where s is the semiperimeter of the triangle:
Heron's formula can also be written as:
Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.
History
The formula is credited to Heron (or Hero) 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 over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.[2]
A formula equivalent to Heron's namely:
- , where
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.
Proof
A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have
by the law of cosines. From this proof get the algebraic statement:
The altitude of the triangle on base a has length b·sin(C), and it follows
The difference of two squares factorization was used in two different steps.
Proof using the Pythagorean theorem
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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 [2]. The following argument reduces Heron's formula directly to the Pythagorean theorem using only elementary means.
We wish to prove 4T2 = 4s(s - a)(s - b)(s - c). The left-hand side equals
while the right-hand side equals
via the identity (p + q)2 - (p - q)2 = 4pq. It therefore suffices to show
and
Substituting 2s = (a + b + c) into the former,
as desired. Similarly, the latter expression becomes
Using the Pythagorean theorem twice, b2 = d2 + h2 and a2 = (c - d)2 + h2, which allows us to simplify the expression to
The result follows.
Numerical stability
Heron's formula as given above is numerically unstable for triangles with a very small angle. A stable alternative [3] [4] involves arranging the lengths of the sides so that and computing
The brackets 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. Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a quadrilateral. Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero.
Heron's formula is also a special case of the formula of the area of a trapezoid based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices,
illustrates its similarity to Tartaglia's formula for the volume of a three-simplex.
Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins.[5]
Heron-type formula for the volume of a tetrahedron
If U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; u opposite to U and so on), then[6]
where
See also
Notes
- ^ "Fórmula de Herón para calcular el área de cualquier triángulo" (in Spanish). Retrieved 30 June 2012.
- ^ Weisstein, Eric W. "Heron's Formula". MathWorld.
- ^ P. Sterbenz (1973). Floating-Point Computation, Prentice-Hall.
- ^ W. Kahan (24 March 2000). "Miscalculating Area and Angles of a Needle-like Triangle" (PDF).
- ^ D. P. Robbins, "Areas of Polygons Inscribed in a Circle", Discr. Comput. Geom. 12, 223-236, 1994.
- ^ W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", [1], pp. 16-17.
References
- Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press. pp. 321–323.