$Any Quadrilateral$

Any Quadrilateral

The area of any quadrilateral from the diagonals and angle, or from the sides with a diagonal or angles.

What to find:

From the diagonals and the angle between them

From the sides and a diagonal

From the sides and the sum of opposite angles

Cyclic, from the sides

The sides go in order: a and b on one side of diagonal d₁, c and d on the other. Four sides without an angle or a diagonal do not determine the area uniquely.

This calculator finds the area of any quadrilateral from the diagonals and angle, from the sides with a diagonal or angles, and the area of a cyclic quadrilateral.

Formulas

S = ½ · d₁ · d₂ · sin φ

S = √((p − a)(p − b)(p − c)(p − d) − abcd · cos²(θ/2))

The second is Bretschneider’s formula. Here p is the semiperimeter and θ is the sum of two opposite angles. For a cyclic quadrilateral θ = 180° and the cosine vanishes, which gives Brahmagupta’s formula.

Example. Diagonals 8 and 6 at 60°: S = ½ · 8 · 6 · sin 60° ≈ 20.78.

Four sides alone are not enough for the area, because a quadrilateral with those sides can flex into different shapes.