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 ...

, Bretschneider's formula is a mathematical expression for the

area Area is the measure of a region's size on a 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 open surface or the boundary of a three-di ...

of a general

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 ...

. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided that directed angles are used.

History

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

Formulation

Bretschneider's formula is expressed as: :

K = \sqrt

= \sqrt .

Here, , , , are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, since

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

as long as directed angles are used so that

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

\alpha+\beta+\gamma+\delta=720^

(when the quadrilateral is crossed).

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 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 ...

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, 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

The second form is given by using the cosine half-angle identity :

\cos^2 \left(\frac\right) = \frac ,

yielding :

K = \sqrt .

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

Related formulae

Bretschneider's formula generalizes 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 ...

, 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 Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

for the area of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. 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 \\
 &=\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

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