Quadrilateral Area Calculator: Any Four-Sided Shape from Sides, Angles or Diagonals

How it works

Bretschneider: Area = √((s − a)(s − b)(s − c)(s − d) − abcd·cos²((A + C)/2)), s = (a + b + c + d)/2 · 4 sides + angle A: q² = a² + d² − 2ad·cos A, cos C = (b² + c² − q²)/(2bc), Area = ½ad·sin A + ½bc·sin C · sides + diagonals (Coolidge): Area = ¼√(4p²q² − (b² + d² − a² − c²)²) · diagonals: Area = ½pq·sin θ · Brahmagupta (cyclic): Area = √((s − a)(s − b)(s − c)(s − d))

Four side lengths alone do not fix a quadrilateral: you can push a four-bar frame into many shapes with different areas. You need one more measurement, and this calculator accepts whichever you have. With two opposite angles it uses Bretschneider’s formula, which generalises Heron’s formula for triangles. With one angle it splits the shape along diagonal BD into two triangles, finds the second diagonal and all four angles, and adds the triangle areas. With both diagonals it uses Coolidge’s side-and-diagonal form; with only the diagonals and the angle where they cross, the area is ½·p·q·sin θ. If the shape is cyclic (its corners lie on a circle, as for a rectangle or an isosceles trapezoid), Brahmagupta’s formula needs only the sides, and the result is the largest area those four sides can enclose. Label the corners A, B, C, D in order around the shape, so a = AB, b = BC, c = CD, d = DA, angle A is between d and a, and C is the opposite corner. Six measurements of a four-sided shape over-determine it, so the calculator checks whether they agree (for example, whether both triangles give the same diagonal) and warns you if one is off. Results come in your length unit plus square meters, square feet, square yards, acres and hectares, which suits irregular plots of land. If you have corner coordinates instead, use the shoelace polygon area calculator; for a trapezoid, rhombus, kite or parallelogram the dedicated calculators are simpler.

Worked example

The default lot has sides a = 120 ft, b = 150 ft, c = 80 ft and d = 50 ft, with a right angle at A and 60° at the opposite corner C. The semiperimeter is s = 400 ÷ 2 = 200, so (s − a)(s − b)(s − c)(s − d) = 80 × 50 × 120 × 150 = 72,000,000. Next abcd·cos²((90° + 60°)/2) = 72,000,000 × cos²75° = 72,000,000 × 0.066987 = 4,823,085. The area is √(72,000,000 − 4,823,085) = √67,176,915 = 8,196.15 ft² (761.45 m², or 0.19 acres), and the perimeter is 400 ft. Both angles give the same diagonal BD = 130 ft, so the measurements are consistent.

Sources: Wikipedia: Bretschneider’s formula (trigonometric and Coolidge diagonal forms) · Omni Calculator: Quadrilateral area (350/120/280/140 ft example) · PlanetCalc: Area of a quadrilateral (sides + angles, sides + diagonals, Brahmagupta)

Frequently asked questions