geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Bretschneider's formula is the following expression for the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

of a general

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

: :

K = \sqrt

= \sqrt .

Here, , , , are the sides of the quadrilateral, is the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...

, and and are any two opposite angles, since

\cos (\alpha+ \gamma) = \cos (\beta+ \delta)

as long as

\alpha+\beta+\gamma+\delta=360^.

Bretschneider's formula works on both convex and concave quadrilaterals (but not crossed ones), whether it is

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

or not. The German mathematician

Carl Anton Bretschneider Carl Anton Bretschneider (27 May 1808 – 6 November 1878) was a mathematician from Gotha, Germany. Bretschneider worked in geometry, number theory, and history of geometry. He also worked on logarithmic integrals and mathematical tables. He was ...

discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt.

Proof

Denote the area of the quadrilateral by . Then we have :

\begin K &= \frac + \frac.\end

Therefore :

2K= (ad) \sin \alpha + (bc) \sin \gamma.

4K^2 = (ad)^2 \sin^2 \alpha + (bc)^2 \sin^2 \gamma + 2abcd \sin \alpha \sin \gamma.

The

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

implies that :

a^2 + d^2 -2ad \cos \alpha = b^2 + c^2 -2bc \cos \gamma,

because both sides equal the square of the length of the diagonal . This can be rewritten as :

\frac = (ad)^2 \cos^2 \alpha +(bc)^2 \cos^2 \gamma -2 abcd \cos \alpha \cos \gamma.

Adding this to the above formula for yields :

\begin 
4K^2 + \frac &= (ad)^2 + (bc)^2 - 2abcd \cos (\alpha + \gamma) \\
                                           &= (ad+bc)^2-2abcd-2abcd\cos(\alpha+\gamma) \\
                                           &= (ad+bc)^2 - 2abcd(\cos(\alpha+\gamma)+1) \\
                                           &= (ad+bc)^2 - 4abcd\left(\frac\right) \\
                                           &= (ad + bc)^2 - 4abcd \cos^2 \left(\frac\right). 
\end

Note that:

\cos^2\frac = \frac

(a trigonometric identity true for all

\frac

) Following the same steps as in

Brahmagupta's formula In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic ...

, this can be written as :

16K^2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) - 16abcd \cos^2 \left(\frac\right).

Introducing the semiperimeter :

s = \frac,

the above becomes :

16K^2 = 16(s-d)(s-c)(s-b)(s-a) - 16abcd \cos^2 \left(\frac\right)

K^2 = (s-a)(s-b)(s-c)(s-d) - abcd \cos^2 \left(\frac\right)

and Bretschneider's formula follows after taking the square root of both sides: :

K = \sqrt

Emmanuel García has used the generalized half angle formulas to give an alternative proof.

Related formulae

Bretschneider's formula generalizes

for the area of a

cyclic quadrilateral 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 ''c ...

, which in turn generalizes

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...

for the area of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals and to give :

\begin
K &=\tfrac\sqrt \\
  &=\sqrt.
\end

Notes

References & further reading

* * C. A. Bretschneider. ''Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes.'' Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261
online copy, German
* F. Strehlke: ''Zwei neue Sätze vom ebenen und sphärischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes''. Archiv der Mathematik und Physik, Band 2, 1842, S. 323-326
online copy, German

External links

* {{MathWorld, urlname=BretschneidersFormula, title=Bretschneider's formula
Bretschneider's formula
at proofwiki.org Theorems about quadrilaterals Area Articles containing proofs