Here is a fast, free (easy) way to get what you want:
Download and install the V4 evaluation version of Rhinoceros.
http://www.rhino3d.com/
Export your graphics from your layout program as simple closed paths in the ai format (. ai v3 always works).
Start Rhino
Start a new File and select the same units you drew your graphics in (i.e. inches mm etc)
Import the ai path file.
Select the paths so they turn yellow.
Go to the "Surface" drop down tab select "Planar curves"
This will turn your outlines into surfaces.
Delete the curves (they will still be selected - just hit delete)
Select all the surfaces
Go to the "Analyse" drop down tab > select "Mass Properties" > "Area"
Look at command line at the top of your workview for the "Cumulative Area"
Done.
This is windows only and the eval never times out.
Update
Opps, its been a while from the last time I did this.
Just checked and you can skip the generating surfaces part.
Rhino will calculate the area of closed curves.
Just import, select and analyse the curves - even easier than i figured.
whk