Heron's formula
In geometry, Heron's formula (or Hero's formula) gives the area A of a triangle in terms of the three side lengths a, b, c. If is the semiperimeter of the triangle, the area is,[1]
It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica, though it was probably known centuries earlier.
Example[edit]
Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. This triangle’s semiperimeter is
and so the area is
In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers.
Alternate expressions[edit]
Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways,
After expansion, the expression under the square root is a quadratic polynomial of the squared side lengths a2, b2, c2.
The same relation can be expressed using the Cayley–Menger determinant,
History[edit]
The formula is credited to Heron (or Hero) of Alexandria (fl. 60 AD),[2] and a proof can be found in his book Metrica. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier,[3] 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.[4]
A formula equivalent to Heron's, namely,
was discovered by the Chinese. It was published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247).[5]
Proofs[edit]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle,[6] or as a special case of De Gua's theorem (for the particular case of acute triangles).[7]
Trigonometric proof using the law of cosines[edit]
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows.[8] 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
Algebraic proof using the Pythagorean theorem[edit]
The following proof is very similar to one given by Raifaizen.[9] 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 cotangents[edit]

If r is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude r and bases a, b, and c. Their combined area is
where is the semiperimeter.
The triangle can alternately be broken into six triangles (in congruent pairs) of altitude r and bases s − a, s − b, and s − c, of combined area (see law of cotangents)
The middle step above is the triple cotangent identity, which applies because the sum of half-angles is
Combining the two, we get
from which the result follows.
Numerical stability[edit]
Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating-point arithmetic. A stable alternative[10][11] 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.
Similar triangle-area formulae[edit]
Three other formulae for the area of a general triangle have a similar structure as Heron's formula, expressed in terms of different variables.
First, if ma, mb, and mc are the medians from sides a, b, and c respectively, and their semi-sum is then[12]
Next, if ha, hb, and hc are the altitudes from sides a, b, and c respectively, and semi-sum of their reciprocals is then[13]
Finally, if α, β, and γ are the three angle measures of the triangle, and the semi-sum of their sines is then[14][15]
where D is the diameter of the circumcircle, This last formula coincides with the standard Heron formula when the circumcircle has unit diameter.
Generalizations[edit]
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.
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.[16]
Heron-type formula for the volume of a tetrahedron[edit]
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[17]
where
Heron formulae in non-Euclidean geometries[edit]
There are also formulae for the area of a triangle in terms of its side lengths for triangles in the sphere or the hyperbolic plane. [18] For a triangle in the sphere with side lengths , half perimeter and area such a formula is
See also[edit]
References[edit]
- ^ 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.
- ^ 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. (2009). "A Heron-type area formula in terms of sines". Mathematical Gazette. 93: 108–109. doi:10.1017/S002555720018430X. S2CID 132042882.
- ^ Kocik, Jerzy; Solecki, Andrzej (2009). "Disentangling a triangle" (PDF). American Mathematical Monthly. 116 (3): 228–237. doi:10.1080/00029890.2009.11920932. S2CID 28155804.
- ^ 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.
- ^ Page 66 in Alekseevskij, D. V.; Vinberg, E. B.; Solodovnikov, A. S. (1993), "Geometry of spaces of constant curvature", in Gamkrelidze, R. V.; Vinberg, E. B. (eds.), Geometry. II: Spaces of constant curvature, Encycl. Math. Sci., vol. 29, Springer-Verlag, pp. 1–138, ISBN 1-56085-072-8
External links[edit]
- 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