I have 4 points on ground, something like what is depicted in this sample:
https://www.mathopenref.com/coordpolygonarea.html
The problem is that I don't know how to determine the coordinates of my points in order for the method described in the provided example to work, because I have only the distances between the couples of points 1 and 2, 2 and 3, 3 and 4, 4 and 1.
Is there some generic formula that gives me the sought for coordinates if I put inside it the 4 distances I know? If so I'll be able to get area of polygon.
I don't need this for more than 4 points.
No, you need to know more information about the points, like the angles between the edges or all pairwise distances. Think of a square that has distances 1 between all 4 points. It has area 1 but you can easily distort it maintaining the same distance between any two neighboring points while its area becomes close to 0.
If it was a triangle you could get its area from the distances using Heron's formula (https://www.mathopenref.com/heronsformula.html). For a 4 -point polygon you need one additional distance, say 1-3. If you have that you can calculate the area of the two triangles given by points 123 and 134.