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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Heron's formula (or Hero's formula) gives the
area of a triangle 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 T=bh/2, where ''b'' is the length of the ''base'' of the triangle, and ''h'' is the ' ...
in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. 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

Let be the triangle with sides , , and . This triangle's semiperimeter is s = \tfrac12(a+b+c)= \tfrac12(4+13+15) = 16 therefore , , , and the area is \begin A &= \\ mu&= \\ mu&= 24. \end In this example, the triangle's side lengths and area are
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. As long as they obey the strict
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
, they define a triangle in the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
whose area is a positive real number.


Alternate expressions

Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways, \begin A &=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt. \end After expansion, the expression under the square root is a
quadratic polynomial In mathematics, a quadratic function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...
of the squared side lengths , , . The same relation can be expressed using the
Cayley–Menger determinant In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its ...
, -16A^2 = \begin 0 & a^2 & b^2 & 1 \\ a^2 & 0 & c^2 & 1 \\ b^2 & c^2 & 0 & 1 \\ 1 & 1 & 1 & 0 \end.


History

The formula is credited to Heron (or Hero) of Alexandria ( 60 AD), and a proof can be found in his book ''Metrica''. Mathematical historian Thomas Heath suggested that
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
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 was discovered by Chinese mathematician Qin Jiushao: A = \frac1\sqrt, published in ''
Mathematical Treatise in Nine Sections The ''Mathematical Treatise in Nine Sections'' () is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of th ...
'' (
Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name Daogu (道古), was a Chinese mathematician, meteorologist, inventor, politician, and writer. He is credited for discovering Horner's method as well as inventing Tianchi basins, a type of rain gau ...
, 1247).


Proofs

There are many ways to prove Heron's formula, for example using
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
as below, or the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles), or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).


Trigonometric proof using the law of cosines

A modern proof, which uses
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and is quite different from the one provided by Heron, follows. Let be the sides of the triangle and the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s opposite those sides. Applying the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
we get \cos \gamma = \frac From this proof, we get the algebraic statement that \sin \gamma = \sqrt = \frac. The
altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...
of the triangle on base has length , and it follows \begin A &= \tfrac12 (\mbox) (\mbox) \\ mu&= \tfrac12 ab\sin \gamma \\ mu&= \frac\sqrt \\ mu&= \tfrac14\sqrt \\ mu&= \tfrac14\sqrt \\ mu&= \sqrt \\ mu&= \sqrt. \end


Algebraic proof using the Pythagorean theorem

The following proof is very similar to one given by Raifaizen. By the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
we have b^2 = h^2 + d^2 and a^2 = h^2 + (c - d)^2 according to the figure at the right. Subtracting these yields a^2 - b^2 = c^2 - 2cd. This equation allows us to express in terms of the sides of the triangle: d = \frac. For the height of the triangle we have that h^2 = b^2 - d^2. By replacing with the formula given above and applying the
difference of squares In elementary algebra, a difference of two squares is one square (algebra), squared number (the number multiplied by itself) subtraction, subtracted from another squared number. Every difference of squares may be Factorization, factored as the Mult ...
identity we get \begin h^2 &= b^2-\left(\frac\right)^2 \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac \\ &= \frac. \end We now apply this result to the formula that calculates the area of a triangle from its height: \begin A &= \frac \\ &= \sqrt \\ &= \sqrt. \end


Trigonometric proof using the law of cotangents

If is the radius of the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...
of the triangle, then the triangle can be broken into three triangles of equal altitude and bases and Their combined area is A = \tfrac12ar + \tfrac12br + \tfrac12cr = rs, where s = \tfrac12(a + b + c) 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) \begin A &= r(s-a) + r(s-b) + r(s-c) \\ mu &= r^2\left(\frac + \frac + \frac\right) \\ mu &= r^2\left(\cot + \cot + \cot\right) \\ mu &= r^2\left(\cot \cot \cot\right)\\ mu &= r^2\left(\frac \cdot \frac \cdot \frac\right) \\ mu &= \frac. \end The middle step above is \cot + \cot + \cot = \cot\cot\cot, the triple cotangent identity, which applies because the sum of half-angles is \tfrac\alpha2 + \tfrac\beta2 + \tfrac\gamma2 = \tfrac\pi2. Combining the two, we get A^2 = s(s - a)(s - b)(s - c), from which the result follows.


Numerical stability

Heron's formula as given above is numerically unstable for triangles with a very small angle when using
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
. A stable alternative involves arranging the lengths of the sides so that a \ge b \ge c and computing A = \tfrac14 \sqrt. The extra brackets indicate the order of operations required to achieve numerical stability in the evaluation.


Similar triangle-area formulae

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
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
s from sides and respectively, and their semi-sum is \sigma = \tfrac12(m_a + m_b + m_c), then A = \frac \sqrt. Next, if , , and are the
altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...
s from sides and respectively, and semi-sum of their reciprocals is H = \tfrac12\bigl(h_a^ + h_b^ + h_c^\bigr), then A^ = 4 \sqrt. Finally, if and are the three angle measures of the triangle, and the semi-sum of their
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
s is S = \tfrac12(\sin\alpha + \sin\beta + \sin\gamma), then \begin A &= D^ \sqrt \\ mu &= \tfrac12 D^ \sin \alpha\,\sin \beta\,\sin \gamma, \end where is the diameter of the
circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...
, D = a/ = b/ = c/. This last formula coincides with the standard Heron formula when the circumcircle has unit diameter.


Generalizations

Heron's formula is a special case of Brahmagupta's formula for the area of a
cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
. Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
. 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 In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
whose sides have lengths as K=\sqrt where s = \tfrac12(a + b + c + d) is the semiperimeter. Heron's formula is also a special case of the
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
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 linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its ...
in terms of the squares of the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
s between the three given vertices, A = \frac \sqrt illustrates its similarity to Tartaglia's formula for the
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of a three-simplex. Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins.


Degenerate and imaginary triangles

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, the semiperimeter will equal the longest side, causing Heron's formula to equal zero. If one of three given lengths is greater than the sum of the other two, then they violate the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
and do not describe the sides of a Euclidean triangle. In this case, Heron's formula gives an imaginary result. For example if and , then . 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.


Volume of a tetrahedron

If are lengths of edges of the tetrahedron (first three form a triangle; opposite to and so on), then \text = \frac where \begin a &= \sqrt \\ b &= \sqrt \\ c &= \sqrt \\ d &= \sqrt \\ X &= (w - U + v)\,(U + v + w) \\ x &= (U - v + w)\,(v - w + U) \\ Y &= (u - V + w)\,(V + w + u) \\ y &= (V - w + u)\,(w - u + V) \\ Z &= (v - W + u)\,(W + u + v) \\ z &= (W - u + v)\,(u - v + W). \end


Spherical and hyperbolic geometry

L'Huilier's formula relates the area of a triangle in
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...
to its side lengths. For a
spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
with side lengths and , semiperimeter , and area , \tan^2 \frac S 4 = \tan \frac s2 \tan\frac2 \tan\frac2 \tan\frac2 For a triangle in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
the analogous formula is \tan^2 \frac S 4 = \tanh \frac s2 \tanh\frac2 \tanh\frac2 \tanh\frac2.


See also

*
Shoelace formula The shoelace formula, also known as Gauss's area formula and the surveyor's formula, is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the ...


Notes and references


External links


A Proof of the Pythagorean Theorem From Heron's Formula
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

Interactive applet and area calculator using Heron's Formula

J. H. Conway discussion on Heron's Formula
*


An alternative proof of Heron's Formula without words

Factoring Heron
{{DEFAULTSORT:Heron's Formula Theorems about triangles Articles containing proofs Area Greek mathematics