Bretschneider's formula

In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals (but not crossed ones), whether it is cyclic or not.

History[edit]

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[edit]

Bretschneider's formula is expressed as:

${\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}}}$

${\displaystyle ={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd[1+\cos(\alpha +\gamma )]}}.}$

Here, a, b, c, d are the sides of the quadrilateral, s is the semiperimeter, and α and γ are any two opposite angles, since ${\displaystyle \cos(\alpha +\gamma )=\cos(\beta +\delta )}$ as long as ${\displaystyle \alpha +\beta +\gamma +\delta =360^{\circ }.}$

Proof[edit]

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

${\displaystyle {\begin{aligned}K&={\frac {ad\sin \alpha }{2}}+{\frac {bc\sin \gamma }{2}}.\end{aligned}}}$

Therefore

${\displaystyle 2K=(ad)\sin \alpha +(bc)\sin \gamma .}$

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

The law of cosines implies that

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

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

${\displaystyle {\frac {(a^{2}+d^{2}-b^{2}-c^{2})^{2}}{4}}=(ad)^{2}\cos ^{2}\alpha +(bc)^{2}\cos ^{2}\gamma -2abcd\cos \alpha \cos \gamma .}$

Adding this to the above formula for 4K² yields

${\displaystyle {\begin{aligned}4K^{2}+{\frac {(a^{2}+d^{2}-b^{2}-c^{2})^{2}}{4}}&=(ad)^{2}+(bc)^{2}-2abcd\cos(\alpha +\gamma )\\&=(ad+bc)^{2}-2abcd-2abcd\cos(\alpha +\gamma )\\&=(ad+bc)^{2}-2abcd(\cos(\alpha +\gamma )+1)\\&=(ad+bc)^{2}-4abcd\left({\frac {\cos(\alpha +\gamma )+1}{2}}\right)\\&=(ad+bc)^{2}-4abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right).\end{aligned}}}$

Note that: ${\displaystyle \cos ^{2}{\frac {\alpha +\gamma }{2}}={\frac {1+\cos(\alpha +\gamma )}{2}}}$ (a trigonometric identity true for all ${\displaystyle {\frac {\alpha +\gamma }{2}}}$)

Following the same steps as in Brahmagupta's formula, this can be written as

${\displaystyle 16K^{2}=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)-16abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right).}$

Introducing the semiperimeter

${\displaystyle s={\frac {a+b+c+d}{2}},}$

the above becomes

${\displaystyle 16K^{2}=16(s-d)(s-c)(s-b)(s-a)-16abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}$

${\displaystyle K^{2}=(s-a)(s-b)(s-c)(s-d)-abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}$

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

${\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}}}$

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

${\displaystyle \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)={\frac {1+\cos \left(\alpha +\gamma \right)}{2}},}$

yielding

${\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd[1+\cos(\alpha +\gamma )]}}.}$

Emmanuel García has used the generalized half angle formulas to give an alternative proof. ^[1]

Related formulae[edit]

Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.

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 e and f to give^[2][3]

${\displaystyle {\begin{aligned}K&={\tfrac {1}{4}}{\sqrt {4e^{2}f^{2}-(b^{2}+d^{2}-a^{2}-c^{2})^{2}}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}((ac+bd)^{2}-e^{2}f^{2})}}.\end{aligned}}}$

Notes[edit]

References & further reading[edit]

• Ayoub, Ayoub B. (2007). "Generalizations of Ptolemy and Brahmagupta Theorems". Mathematics and Computer Education. 41 (1). ISSN 0730-8639.
• 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[edit]

• Weisstein, Eric W. "Bretschneider's formula". MathWorld.
• Bretschneider's formula at proofwiki.org