Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of minimal surfaces and the Plateau problem there exists a surface of minimal area with this lines as boundary. The situation looks like this:

(Picture from http://mathworld.wolfram.com/SkewQuadrilateral.html) but note that the obvious bilinear interpolation is not the minimal surface.

There are formulae for the minimal surfaces such as the Weierstraß-Enneper formula but I haven't come across a formula for this particular case of a quadrilateral.

In fact, I am not interested in a formula for the surface but only look for an answer to the question:

What is area of the minimal surface of the quadrilateral in terms of the four corner points $x_1,x_2,x_3,x_4$?