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There are several ways to calculate area of a quadrilateral

Picture: wikipedia

1. Area of a quadrilateral given diagonals and angle between them. In this case formula will be
$S=\frac{d_1d_2}{2}sin\theta$

Calculator:

°
Digits after the decimal point: 2
Area

1. Area of a quadrilateral given four sides and two opposite angles. In this case it will be Bretschneider's formula

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

Calculator:

Area of a general quadrilateral given four sides and two opposite angles

°
°
Digits after the decimal point: 2
Area

1. Area of a quadrilateral given four sides and two diagonals. In this case it will be non-trigonometric Bretschneider's formula

$F={\tfrac {1}{4}}{\sqrt {4e^{2}f^{2}-(b^{2}+d^{2}-a^{2}-c^{2})^{2}}}\\={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+ef)(ac+bd-ef)}}$,
where s - semiperimeter

Calculator:

Area of a general quadrilateral given four sides and two diagonals.

Digits after the decimal point: 2
Area

1. Area of a quadrilateral given four sides and the fact that it is cyclic quadrilateral. This is special case of Bretschneider's formula (we know that sum of two opposite angles are 180) known as Brahmagupta's formula

$K={\sqrt {(s-a)(s-b)(s-c)(s-d)}}$,
where s - semiperimeter

For this you can use calculator above by entering arbitrary angles those sum is 180.

Prove for Bretschneider's formulas can be found here

And it is worth to mention that it is impossible to find area of quadrilateral given ONLY four sides. You will need some additional conditions anyway, like those above. Yet there were many people on the site asking for this, so we've done joke calculator which calculates area of quadrilateral by four sides - for endless number of quadrilaterals. You can find it at Area of an irregular quadrangle with the given sides.