Heron's formula

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A triangle with sides a, b, and c

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]

Triangle with altitude h cutting base c into d + (cd)

The following proof is very similar to one given by Raifaizen.[9] By the Pythagorean theorem we have b2 = h2 + d2 and a2 = h2 + (cd)2 according to the figure at the right. Subtracting these yields a2b2 = 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 = b2d2. 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]

Geometrical significance of sa, sb, and sc. See the law of cotangents for the reasoning behind this.

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 sa, sb, and sc, 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 abc 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]

Cyclic Quadrilateral

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

while for the hyperbolic plane we have

See also[edit]

References[edit]

  1. ^ 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.
  2. ^ 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.
  3. ^ Heath, Thomas L. (1921). A History of Greek Mathematics. Vol. II. Oxford University Press. pp. 321–323.
  4. ^ Weisstein, Eric W. "Heron's Formula". MathWorld.
  5. ^ 秦, 九韶 (1773). "卷三上, 三斜求积". 數學九章 (四庫全書本) (in Chinese).
  6. ^ "Personal email communication between mathematicians John Conway and Peter Doyle". 15 December 1997. Retrieved 25 September 2020.
  7. ^ 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.
  8. ^ Niven, Ivan (1981). Maxima and Minima Without Calculus. The Mathematical Association of America. pp. 7–8.
  9. ^ Raifaizen, Claude H. (1971). "A Simpler Proof of Heron's Formula". Mathematics Magazine. 44 (1): 27–28. doi:10.1080/0025570X.1971.11976093.
  10. ^ 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.
  11. ^ William M. Kahan (24 March 2000). "Miscalculating Area and Angles of a Needle-like Triangle" (PDF).
  12. ^ Benyi, Arpad, "A Heron-type formula for the triangle," Mathematical Gazette 87, July 2003, 324–326.
  13. ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
  14. ^ Mitchell, Douglas W. (2009). "A Heron-type area formula in terms of sines". Mathematical Gazette. 93: 108–109. doi:10.1017/S002555720018430X. S2CID 132042882.
  15. ^ Kocik, Jerzy; Solecki, Andrzej (2009). "Disentangling a triangle" (PDF). American Mathematical Monthly. 116 (3): 228–237. doi:10.1080/00029890.2009.11920932. S2CID 28155804.
  16. ^ D. P. Robbins, "Areas of Polygons Inscribed in a Circle", Discr. Comput. Geom. 12, 223-236, 1994.
  17. ^ W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", [1], pp. 16–17.
  18. ^ 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

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