How to calculate area of this shape? 
I was trying to solve a complicated problem then I came accros to this complicated problem. I believe that there is enough information to calculate the area. Can you help me to find a general formula for the area of this shape, in terms of $x,\alpha,\beta$?
I forgot to write on the figure: $|AB|$ is tilted $45^\circ$ w.r.t. "ground", $\beta<\alpha$ and $|AB|$ is not parallel to $|DC|$. 
$|CB|=|DA|=1$ unit and $|AB|=x$.

Set up a coordinate system with its original at $B$. Then, in this coordinate system, we have:
$$
A = (x_0, y_0) = \frac{1}{\sqrt2}(x,x)
$$
$$
B = (x_1, y_1) = (0,0)
$$
$$
C = (x_2,y_2) = (-\cos\beta, \sin\beta)
$$
$$
D = (x_3,y_3) = \frac{1}{\sqrt2}(x,x) + (-\cos\alpha, \sin\alpha)
$$
Then apply the area formula from here:
$$
\text{Area} = \frac12\sum_{i=0}^{n-1}(x_iy_{i+1} -x_{i+1}y_i)
$$
