Editing Quadrilateral (section)

===Trigonometric formulas===
The area can be expressed in trigonometric terms as<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Quadrilateral|url=https://mathworld.wolfram.com/Quadrilateral.html|access-date=2020-09-02|website=mathworld.wolfram.com|language=en}}</ref>
:<math>K = \tfrac12 pq \sin \theta,</math>

where the lengths of the diagonals are {{math|''p''}} and {{math|''q''}} and the angle between them is {{math|''θ''}}.<ref>Harries, J. "Area of a quadrilateral," ''Mathematical Gazette'' 86, July 2002, 310–311.</ref> In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to <math>K=\tfrac{pq}{2}</math> since {{math|''θ''}} is {{math|90°}}.

The area can be also expressed in terms of bimedians as<ref name=Josefsson4/>
:<math>K = mn \sin \varphi,</math>
where the lengths of the bimedians are {{math|''m''}} and {{math|''n''}} and the angle between them is {{math|''φ''}}.

[[Bretschneider's formula]]<ref>R. A. Johnson, ''Advanced Euclidean Geometry'', 2007, [[Dover Publications|Dover Publ.]], p. 82.</ref><ref name=":1" /> expresses the area in terms of the sides and two opposite angles:
:<math>\begin{align}
K &= \sqrt{(s-a)(s-b)(s-c)(s-d) - \tfrac{1}{2} abcd \; [ 1 + \cos (A + C) ]} \\
&= \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd\, \cos^2 \tfrac12(A + C) }
\end{align}</math>

where the sides in sequence are {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}}, where {{math|''s''}} is the semiperimeter, and {{math|''A''}} and {{math|''C''}} are two (in fact, any two) opposite angles. This reduces to [[Brahmagupta's formula]] for the area of a cyclic quadrilateral—when {{math|{{nobreak|''A'' + ''C'' {{=}} 180°}} }}.

Another area formula in terms of the sides and angles, with angle {{math|''C''}} being between sides {{math|''b''}} and {{math|''c''}}, and {{math|''A''}} being between sides {{math|''a''}} and {{math|''d''}}, is
:<math>K = \tfrac12 ad \sin{A} + \tfrac12 bc \sin{C}.</math>

In the case of a cyclic quadrilateral, the latter formula becomes <math>K = \tfrac12(ad+bc)\sin{A}.</math>

In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to <math>K=ab \cdot \sin{A}.</math>

Alternatively, we can write the area in terms of the sides and the intersection angle {{math|''θ''}} of the diagonals, as long as {{math|''θ''}} is not {{math|90°}}:<ref name=Mitchell>Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306–309.</ref>
:<math>K = \tfrac14 \left|\tan \theta\right| \cdot \left| a^2 + c^2 - b^2 - d^2 \right|.</math>

In the case of a parallelogram, the latter formula becomes <math>K = \tfrac12 \left|\tan \theta\right| \cdot \left| a^2 - b^2 \right|.</math>

Another area formula including the sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} is<ref name=Josefsson4>{{citation
 | last = Josefsson
 | first = Martin
 | journal = Forum Geometricorum
 | pages = 17–21
 | title = Five Proofs of an Area Characterization of Rectangles
 | url = http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf
 | volume = 13
 | year = 2013
 | access-date = 2013-02-20
 | archive-date = 2016-03-04
 | archive-url = https://web.archive.org/web/20160304001152/http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf
 | url-status = dead
 }}.</ref>
:<math>K=\tfrac12 \sqrt{\bigl((a^2+c^2)-2x^2\bigr)\bigl((b^2+d^2)-2x^2\bigr)} \sin{\varphi}</math>
where {{math|''x''}} is the distance between the midpoints of the diagonals, and {{math|''φ''}} is the angle between the [[Quadrilateral#Special line segments|bimedian]]s.

The last trigonometric area formula including the sides {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} and the angle {{math|''α''}} (between {{math|''a''}} and {{math|''b''}}) is:<ref>{{Cite web |date=2009 |title=Triangle formulae |url=https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-triangleformulae-2009-1.pdf |access-date=26 June 2023 |website=mathcentre.ac.uk}}</ref>
:<math>K=\tfrac12 ab \sin{\alpha}+\tfrac14 \sqrt{4c^2d^2-(c^2+d^2-a^2-b^2+2ab \cos{\alpha})^2} ,</math>
which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle {{math|''α''}}), by just changing the first sign {{math|+}} to {{math|-}}.