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;
ha – 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 quadrilateral can be inscribed in a circle if the sum of its opposite angles is equal to 180°), and a circle can be circumscribed around it, then its area is equal to: