In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be 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.
Let be the triangle with sides and 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.
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 , , .
The same relation can be expressed using the Cayley–Menger determinant, [2]
The formula is credited to Heron (or Hero) of Alexandria (fl. 60 AD), [3] and a proof can be found in his book Metrica. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier, [4] 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. [5]
A formula equivalent to Heron's, namely,
was discovered by the Chinese. It was published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247). [6]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [7] or as a special case of De Gua's theorem (for the particular case of acute triangles), [8] 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. [9] Let be the sides of the triangle and 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 has length , and it follows
The following proof is very similar to one given by Raifaizen. [10] By the Pythagorean theorem we have and according to the figure at the right. Subtracting these yields This equation allows us to express in terms of the sides of the triangle:
For the height of the triangle we have that By replacing 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:
If is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude and bases and Their combined area is
where is the semiperimeter.
The triangle can alternately be broken into six triangles (in congruent pairs) of altitude and bases and 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.
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 and computing [11] [12]
The brackets in the above formula are required in order to prevent numerical instability 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.
First, if and are the medians from sides and respectively, and their semi-sum is then [13]
Next, if , , and are the altitudes from sides and respectively, and semi-sum of their reciprocals is then [14]
Finally, if and are the three angle measures of the triangle, and the semi-sum of their sines is then [15] [16]
where is the diameter of the circumcircle, This last formula coincides with the standard Heron formula when the circumcircle has unit diameter.
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 of a cyclic quadrilateral whose sides have lengths as
where is the semiperimeter.
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. [17]
If are lengths of edges of the tetrahedron (first three form a triangle; opposite to and so on), then [18]
where
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. [19] For a triangle in the sphere with side lengths and the semiperimeter and area , such a formula is
while for the hyperbolic plane we have
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In mechanics and geometry, the 3D rotation group, often denoted O(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi. Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.
In mathematics, the Gaussian or ordinary hypergeometric function2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
In mathematics, the lemniscate constantϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of π for the circle. Equivalently, the perimeter of the lemniscate is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol ϖ is a cursive variant of π; see Pi § Variant pi.
In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not.
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya.
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is where is the volume of the unit n-ball, the n-ball of radius 1.
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle.
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.