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 (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:
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[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
cos
(
α
+
γ
)
=
cos
(
β
+
δ
)
{\displaystyle \cos(\alpha +\gamma )=\cos(\beta +\delta )}
as long as
α
+
β
+
γ
+
δ
=
360
∘
.
{\displaystyle \alpha +\beta +\gamma +\delta =360^{\circ }.}
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 2 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}}}
(a trigonometric identity true for all
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[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]
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
)
.
{\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}}}
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 ]