Given a quadrilateral with sides of length a, b, c, d and opposite interior
angles A and C (it turns out not to matter which two opposite corners are
selected), then its area, K = sqrt[(s-a)(s-b)(s-c)(s-d) - abcd cos²([A+C]/2)].

**Bretschneider's Formula for the area of a quadrilateral**

Let ABCD be a general quadrilateral with sides of length a, b, c, d.
Its two diagonals are p and q, with p = BD, q = AC intersecting at O.

The perimeter, P, is defined as P = a + b + c + d.

The semiperimeter, s, is defined as s = P/2 = (a+b+c+d)/2

Angle between diagonals: θ

A + B + C + D = 2π radians = 360�

K = pq sin(θ)/2

K = (b²+d²-a²-c²)tan(θ)/4

K = sqrt[4p²q²-(b²+d²-a²-c²)²]/4

K = sqrt[(s-a)(s-b)(s-c)(s-d) - abcd cos²([A+C]/2)]

In particular if ABCD is cyclic, [A+C]/2 = 90� and this reduces to the
Brahmagupta formula for the area of a cyclic quadrilateral (of which
Heron's formula for a triangle is a yet-more-special case).

### Internet references:

Dr. Math's
Quadrilateral Formulas has a very quick derivation of the formula, along
with quite a few other quadrilateral formulas.

Mathworld
Bretschneiders Formula has a vector derivation of the formula.

### Related pages in this website:

Summary of geometrical theorems

Law of Cosines

Heron's Formula for
the area of a triangle.

Brahmagupta's
Formula for the area of a cyclic quadrilateral, which can be considered a
generalization of Heron's Formula.

Point and Triangle -- a series
of pages explaining how to determine whether a point is inside a triangle.
One of these pages
gives an easy method using Heron's formula: Point D is inside triangle ABC if
the sums of the areas of ABD, BCD and CAD equal the area of ABC.

Triangle Area using
Determinant

Triangle Area using Vectors

Polygon Area using
Determinant

The webmaster and author of this Math Help site is
Graeme McRae.