Right Quadrilateral Calculator

Name: Right Quadrilateral - Geometry Calculator
Author: Jürgen Kummer

Calculations for a convex right quadrilateral or right quadrangle, a quadrilateral with exactly one right angle.
Enter the lengths of the four sides. The right angle α is located between sides a and d. For the quadrilateral to close, the sum of sides b and c together must be greater than diagonal f, and the difference between them must be less than f. Angles are calculated in degrees, here you can convert angle units.

Shape of the right quadrilateral (a at the bottom, b on the right, c at the top, d on the left):

Formulas:

α = 90 °

f = \sqrt{a^{2} + d^{2}}

β = \arccos (\frac{a^{2} + f^{2} - d^{2}}{2 a f}) + \arccos (\frac{b^{2} + f^{2} - c^{2}}{2 b f})

γ = \arccos (\frac{b^{2} + c^{2} - f^{2}}{2 b c})

δ = 270 ° - β - γ

e = \sqrt{a^{2} + b^{2} - 2 a b \cdot \cos (β)}

p = a + b + c + d

q = f + b + c

A = \frac{ad}{2} + \sqrt{\frac{q}{2} \cdot (\frac{q}{2} - f) \cdot (\frac{q}{2} - b) \cdot (\frac{q}{2} - c)}

Side length, diagonals and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the angles are in degrees.

A quadrilateral with two adjacent right angles is a right trapezoid. With four right angles, it is a rectangle. If there are three right angles, the fourth angle must also be a right angle. Similarly, with two opposite right angles, a quadrilateral necessarily becomes a rectangle. In general, for a quadrilateral with one given angle and fixed, valid side lengths, two of the remaining angles can be chosen independently, while the fourth is determined automatically by the sum of the angles.
The calculation is performed by decomposing the shape into a right triangle with legs a and d and hypotenuse f, and a triangle with sides f, b and c. Convexity is ensured by the chosen arrangement of the two triangles. The auxiliary variable q in the formulas represents the perimeter of triangle fbc. Angle β is the sum of a non-right angle from the right triangle and the adjacent angle from the general triangle. Angle δ could be determined in the same way. However, since calculating γ is simpler using the law of cosines on the general triangle, γ is calculated first, and δ is then derived from the quadrilateral's angle sum of 360 degrees, or 270 degrees for the three non-right angles.
Like the general quadrilateral, this right quadrilateral possesses no regularity or symmetry. However, the presence of a right angle distinguishes it from the general case, because right angles are often important for many applications.

Calculation of area and other dimensions: Right Quadrilateral

Last updated on 08/09/2026. Author: Jürgen Kummer

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