Bretschneider Quadrilateral Area Calculator

How it is calculated

The area uses the semi-perimeter s = (a + b + c + d) ÷ 2 and two opposite angles A and C in degrees: Area = √[(s−a)(s−b)(s−c)(s−d) − a·b·c·d·cos²((A + C) ÷ 2)]. Sides are in any length unit and the area comes out in that unit squared. The tool also shows s and the cosine term cos((A + C) ÷ 2). When A + C = 180° the cosine is 0 and the formula becomes Brahmagupta's formula for a cyclic quadrilateral. Sides must be above zero, and if the expression under the root is negative (the sides and angles cannot form a real quadrilateral) there is no result. The tool does not check anything else: for instance, it does not verify that the angles are the true angles between your sides.

Common mistakes

• Using two adjacent angles instead of two opposite ones. The formula needs the angles at opposite corners (A and C, or B and D); adjacent angles give a wrong area.
• Entering angles in radians. The tool expects degrees, so 1.57 would be read as under 2 degrees.
• Giving sides in an order that does not go around the shape. The sides a, b, c, d must follow the perimeter in sequence, and A, C must sit at the opposite corners that match.
• Inventing an angle that is inconsistent with the sides. The formula does not detect a quadrilateral that cannot exist unless the root becomes negative, so check with a sketch.
• Expecting the angle values to matter when the shape is cyclic. If A + C is 180°, any such pair gives the same area, as Brahmagupta's formula uses only the sides.