I've searched the web now for ages to try and find a paper on the asymptotic distribution of the area of the union of randomly placed discs on the plane. Ideally, I would be looking for the discs to be placed in a unit square, but if anyone knows of other work that has been done, that would be great. I can calculate the expected area, and I have found a number of papers on the topic of 'coverage of wireless sensor networks' which give some useful ideas. Nevertheless, I was wondering whether this problem has ever been studied for its own sake.
If by "the area of ... circles" you mean the area of the union of disks, then there is work in the statistical literature, e.g.:
Zikmundová, Markéta, Kateřina Staňková Helisová, and Viktor Beneš. "Spatio-temporal model for a random set given by a union of interacting discs." Methodology and Computing in Applied Probability 14.3 (2012): 883-894.
Møller, Jesper, and Kateřina Helisová. "Likelihood inference for unions of interacting discs." Scandinavian Journal of Statistics 37.3 (2010): 365-381.
Møller, Jesper, and Kateřina Helisová. "Power diagrams and interaction processes for unions of discs." Advances in Applied Probability (2008): 321-347.
Fig. 1 from 3rd paper above.