But in 2D intersections (being points) don't have areas... at least, normally they don't...
This looks like a 2D question, though. And it's a nice drawing. Is it a puzzle?
Would you like to determine the number of intersections between the rectangles?
Are all other rectangles overlapping rectangle #1?
If you want to find the coords at the intersections of "overlapping" rectangles,
you could use a parametric test between a point and all rectangles that the point does not belong to, and contain the point.
If you're going for the total area, and you know that every bit of the outermost rectangle will be filled,
(max(all_coords[o..n-1].x) - min(all_coords[o..n-1].x)) * (max(all_coords[o..n-1].y) - min(all_coords[o..n-1].y))
This doesn't sum overlapping areas, of course.