Question

If I'm given the four sides of an irregular quadrilateral but none of the angles, is it possible to calculate its area?

Answer 1

No

Explanation:

Quadrilateral cannot be defined only by its sides (unless we know beforehand that this is a rectangle).
For example, make a parallelogram. Without changing its sides we can "fold" it down to a line, thus making its altitude (and, therefore, area) as small as possible.

The same is with any irregular quadrilateral. Without angles defined, it can be transformed into another quadrilateral with different shape and area.