Area of a convex quadrilateral

Formulas for the area of a convex quadrilateral

Designations: a, b, c and d – sides of the quadrilateral;

r - radius of the inscribed circle, R - radius of the circumscribed circle;

h_a – height dropped to the side a;

p – semiperimeter,
d₁, d₂ - diagonals of a quadrilateral.

Area of an arbitrary convex quadrilateral

where φ - the angle between diagonals.

Area of an inscribed quadrilateral

where p - the semiperimeter.

Area of the circumscribed quadrilateral

S = p * r

If a circle can be inscribed in a quadrilateral (a circle can be inscribed in a quadrilateral when the sums of its opposite sides are equal), and a circle can be circumscribed around it (a quadrilateral can be inscribed in a circle if the sum of its opposite angles is equal to 180°), then its area is equal to:

The area of a parallelogram with side a and height h_a dropped to this side

Area of a parallelogram with sides a, b and angle γ between them

Area of a rhombus with diagonals d₁, d₂

Area of a rhombus with side a and angle γ

Area of a rectangle with sides a, b

S = a * b

Area of a square with side a

S = a²

Area of a square with diagonal d

S = 1/2 * d²

Area of a trapezoid with bases a, b and height h

S = 1/2 * (a + b)* h