Heron's formula
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.
However, Heron's formula works equally well when the side lengths are real numbers.
As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways,
The formula is credited to Heron (or Hero) of Alexandria (fl.
60 AD),[4] and a proof can be found in his book Metrica.
Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier,[5] 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.
[6] A formula equivalent to Heron's was discovered by Chinese mathematician Qin Jiushao:
published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247).
[7] There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle,[8] or as a special case of De Gua's theorem (for the particular case of acute triangles),[9] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows.
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:
the triple cotangent identity, which applies because the sum of half-angles is
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
The extra brackets indicate the order of operations required to achieve numerical stability in the evaluation.
Three other formulae for the area of a general triangle have a similar structure as Heron's formula, expressed in terms of different variables.
are the three angle measures of the triangle, and the semi-sum of their sines is
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.
[18] If one of three given lengths is equal to the sum of the other two, the three sides determine a degenerate triangle, a line segment with zero area.
In this case, Heron's formula gives an imaginary result.
This can be interpreted using a triangle in the complex coordinate plane
, where "area" can be a complex-valued quantity, or as a triangle lying in a pseudo-Euclidean plane with one space-like dimension and one time-like dimension.
are lengths of edges of the tetrahedron (first three form a triangle;
L'Huilier's formula relates the area of a triangle in spherical geometry to its side lengths.
For a triangle in hyperbolic geometry the analogous formula is
