(Redirected from Bretschneider's Formula )
Formula for the area of a quadrilateral
A quadrilateral. 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. 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
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
⋅
cos
2
(
α
+
γ
2
)
{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}}}
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
2
a
b
c
d
[
1
+
cos
(
α
+
γ
)
]
.
{\displaystyle ={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd}}.}
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
cos
(
α
+
γ
)
=
cos
(
β
+
δ
)
{\displaystyle \cos(\alpha +\gamma )=\cos(\beta +\delta )}
α
+
β
+
γ
+
δ
=
360
∘
{\displaystyle \alpha +\beta +\gamma +\delta =360^{\circ }}
α
+
β
+
γ
+
δ
=
720
∘
{\displaystyle \alpha +\beta +\gamma +\delta =720^{\circ }}
Proof
Denote the area of the quadrilateral by K . Then we have
K
=
a
d
sin
α
2
+
b
c
sin
γ
2
.
{\displaystyle {\begin{aligned}K&={\frac {ad\sin \alpha }{2}}+{\frac {bc\sin \gamma }{2}}.\end{aligned}}}
Therefore
2
K
=
(
a
d
)
sin
α
+
(
b
c
)
sin
γ
.
{\displaystyle 2K=(ad)\sin \alpha +(bc)\sin \gamma .}
4
K
2
=
(
a
d
)
2
sin
2
α
+
(
b
c
)
2
sin
2
γ
+
2
a
b
c
d
sin
α
sin
γ
.
{\displaystyle 4K^{2}=(ad)^{2}\sin ^{2}\alpha +(bc)^{2}\sin ^{2}\gamma +2abcd\sin \alpha \sin \gamma .}
The law of cosines implies that
a
2
+
d
2
−
2
a
d
cos
α
=
b
2
+
c
2
−
2
b
c
cos
γ
,
{\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
(
a
2
+
d
2
−
b
2
−
c
2
)
2
4
=
(
a
d
)
2
cos
2
α
+
(
b
c
)
2
cos
2
γ
−
2
a
b
c
d
cos
α
cos
γ
.
{\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 4K yields
4
K
2
+
(
a
2
+
d
2
−
b
2
−
c
2
)
2
4
=
(
a
d
)
2
+
(
b
c
)
2
−
2
a
b
c
d
cos
(
α
+
γ
)
=
(
a
d
+
b
c
)
2
−
2
a
b
c
d
−
2
a
b
c
d
cos
(
α
+
γ
)
=
(
a
d
+
b
c
)
2
−
2
a
b
c
d
(
cos
(
α
+
γ
)
+
1
)
=
(
a
d
+
b
c
)
2
−
4
a
b
c
d
(
cos
(
α
+
γ
)
+
1
2
)
=
(
a
d
+
b
c
)
2
−
4
a
b
c
d
cos
2
(
α
+
γ
2
)
.
{\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:
cos
2
α
+
γ
2
=
1
+
cos
(
α
+
γ
)
2
{\displaystyle \cos ^{2}{\frac {\alpha +\gamma }{2}}={\frac {1+\cos(\alpha +\gamma )}{2}}}
α
+
γ
2
{\displaystyle {\frac {\alpha +\gamma }{2}}}
Following the same steps as in Brahmagupta's formula , this can be written as
16
K
2
=
(
a
+
b
+
c
−
d
)
(
a
+
b
−
c
+
d
)
(
a
−
b
+
c
+
d
)
(
−
a
+
b
+
c
+
d
)
−
16
a
b
c
d
cos
2
(
α
+
γ
2
)
.
{\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
s
=
a
+
b
+
c
+
d
2
,
{\displaystyle s={\frac {a+b+c+d}{2}},}
the above becomes
16
K
2
=
16
(
s
−
d
)
(
s
−
c
)
(
s
−
b
)
(
s
−
a
)
−
16
a
b
c
d
cos
2
(
α
+
γ
2
)
{\displaystyle 16K^{2}=16(s-d)(s-c)(s-b)(s-a)-16abcd\cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}
K
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
cos
2
(
α
+
γ
2
)
{\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:
K
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
⋅
cos
2
(
α
+
γ
2
)
{\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
cos
2
(
α
+
γ
2
)
=
1
+
cos
(
α
+
γ
)
2
,
{\displaystyle \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)={\frac {1+\cos \left(\alpha +\gamma \right)}{2}},}
yielding
K
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
2
a
b
c
d
[
1
+
cos
(
α
+
γ
)
]
.
{\displaystyle K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd}}.}
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 , 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
K
=
1
4
4
e
2
f
2
−
(
b
2
+
d
2
−
a
2
−
c
2
)
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
4
(
(
a
c
+
b
d
)
2
−
e
2
f
2
)
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
1
4
(
a
c
+
b
d
+
e
f
)
(
a
c
+
b
d
−
e
f
)
{\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})}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+ef)(ac+bd-ef)}}\\\end{aligned}}}
Notes
Coolidge, J. L. (1939). "A Historically Interesting Formula for the Area of a Quadrilateral". The American Mathematical Monthly . 46 (6): 345–347. doi :10.2307/2302891 . JSTOR 2302891 .
Hobson, E. W. (1918). A Treatise on Plane Trigonometry
References & further reading
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
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↑
as long as directed angles are used so that 
or 
(when the quadrilateral is crossed).
(a trigonometric identity true for all 
)
