In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle when the lengths of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.
Heron's formula can also be written as
Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. This triangle’s semiperimeter is
and so the area is
The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book Metrica, written around AD 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.
A formula equivalent to Heron's, namely,
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles).
Trigonometric proof using the law of cosinesEdit
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows. Let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. Applying the law of cosines we get
From this proof, we get the algebraic statement that
The altitude of the triangle on base a has length b sin γ, and it follows
The difference of two squares factorization was used in two different steps.
Algebraic proof using the Pythagorean theoremEdit
The following proof is very similar to one given by Raifaizen. By the Pythagorean theorem we have b2 = h2 + d2 and a2 = h2 + (c − d)2 according to the figure at the right. Subtracting these yields a2 − b2 = c2 − 2cd. This equation allows us to express d in terms of the sides of the triangle:
For the height of the triangle we have that h2 = b2 − d2. By replacing d with the formula given above and applying the difference of squares identity we get
We now apply this result to the formula that calculates the area of a triangle from its height:
Trigonometric proof using the law of cotangentsEdit
and A = rs, but, since the sum of the half-angles is π/2, the triple cotangent identity applies, so the first of these is
Combining the two, we get
from which the result follows.
Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating-point arithmetic. A stable alternative involves arranging the lengths of the sides so that a ≥ b ≥ c and computing
The brackets in the above formula are required in order to prevent numerical instability in the evaluation.
Other area formulae resembling Heron's formulaEdit
Three other area formulae have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum 1/2(ma + mb + mc) as σ, we have
Next, denoting the altitudes from sides a, b, and c respectively as ha, hb, and hc, and denoting the semi-sum of the reciprocals of the altitudes as H = 1/2(h−1
a + h−1
b + h−1
c) we have
Finally, denoting the semi-sum of the angles' sines as S = 1/2(sin α + sin β + sin γ), we have
where D is the diameter of the circumcircle: D = a/sin α = b/sin β = c/sin γ.
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.
Brahmagupta's formula gives the area K of a cyclic quadrilateral whose sides have lengths a, b, c, d as
where s, the semiperimeter, is defined to be
Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
Heron-type formula for the volume of a tetrahedronEdit
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
- "Fórmula de Herón para calcular el área de cualquier triángulo" (in Spanish). Spain: Ministerio de Educación, Cultura y Deporte. 2004. Retrieved 30 June 2012.
- Kendig, Keith (2000). "Is a 2000-year-old formula still keeping some secrets?". The American Mathematical Monthly. 107 (5): 402–415. doi:10.1080/00029890.2000.12005213. JSTOR 2695295. MR 1763392. S2CID 1214184.
- Id, Yusuf; Kennedy, E. S. (1969). "A medieval proof of Heron's formula". The Mathematics Teacher. 62 (7): 585–587. doi:10.5951/MT.62.7.0585. JSTOR 27958225. MR 0256819.
- Heath, Thomas L. (1921). A History of Greek Mathematics. Vol. II. Oxford University Press. pp. 321–323.
- Weisstein, Eric W. "Heron's Formula". MathWorld.
- 秦, 九韶 (1773). "卷三上, 三斜求积". 數學九章 (四庫全書本) (in Chinese).
- "Personal email communication between mathematicians John Conway and Peter Doyle". 15 December 1997. Retrieved 25 September 2020.
- Lévy-Leblond, Jean-Marc (2020-09-14). "A Symmetric 3D Proof of Heron's Formula". The Mathematical Intelligencer. 43 (2): 37–39. doi:10.1007/s00283-020-09996-8. ISSN 0343-6993.
- Niven, Ivan (1981). Maxima and Minima Without Calculus. The Mathematical Association of America. pp. 7–8.
- Raifaizen, Claude H. (1971). "A Simpler Proof of Heron's Formula". Mathematics Magazine. 44 (1): 27–28. doi:10.1080/0025570X.1971.11976093.
- The second part of the Law of cotangents proof depends on Heron's formula itself, but this article depends only on the first part.
- Sterbenz, Pat H. (1974-05-01). Floating-Point Computation. Prentice-Hall Series in Automatic Computation (1st ed.). Englewood Cliffs, New Jersey, USA: Prentice Hall. ISBN 0-13-322495-3.
- William M. Kahan (24 March 2000). "Miscalculating Area and Angles of a Needle-like Triangle" (PDF).
- Benyi, Arpad, "A Heron-type formula for the triangle," Mathematical Gazette" 87, July 2003, 324–326.
- Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
- Mitchell, Douglas W., "A Heron-type area formula in terms of sines," Mathematical Gazette 93, March 2009, 108–109.
- 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?", , pp. 16–17.
- A Proof of the Pythagorean Theorem From Heron's Formula at cut-the-knot
- Interactive applet and area calculator using Heron's Formula
- J. H. Conway discussion on Heron's Formula
- "Heron's Formula and Brahmagupta's Generalization". MathPages.com.
- A Geometric Proof of Heron's Formula
- An alternative proof of Heron's Formula without words
- Factoring Heron