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)].
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).
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.
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
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