Quadrilateral Area (Bretschneider Formula)

Compute the area of any quadrilateral from its four side lengths and the sum of two opposite angles using the Bretschneider formula. Perimeter and semiperimeter are shown too.

Input

Enter the four side lengths a, b, c, d and the sum of one pair of opposite interior angles (A plus C). The Bretschneider formula gives the area of a general quadrilateral.

Side a

Side b

Side c

Side d

Opposite angle sum A plus C (degrees)

Sum of one pair of opposite interior angles. Use 180 for a cyclic quadrilateral.

Result

Quadrilateral area

24.494897

Perimeter

Semiperimeter s

When all sides use the same unit, the area is given in that unit squared.

How it works

The semiperimeter s is s equals (a plus b plus c plus d) divided by 2.
The area is the square root of (s minus a)(s minus b)(s minus c)(s minus d) minus abcd times cos squared of ((A plus C) divided by 2).
A plus C is the sum of one pair of opposite interior angles in degrees. For a cyclic quadrilateral A plus C equals 180 degrees, and the result matches the Brahmagupta formula.
If the value under the square root becomes negative, the side and angle combination cannot form a real quadrilateral and no area is returned.
Use the same unit for every side length. The area is expressed in that unit squared.

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