Thanks to web tracking data, I see that there are many searches with the key words "measure" "area" and "GIS" or some variation of that theme and sometimes people stumble upon this blog. How to measure area in GIS has been a difficult undertaking, for some odd reason that I haven't yet discovered the answer.
Recently and by pure chance I found that ArcMap DOES in fact measure area very easily. Perhaps it was with ArcGIS 9.2 upgrade, but here is how you do it. I guess if you do not have the 9.2 upgrade then you need to ignore this entry because I don't believe this worked for 9.1 or earlier. As shown in the Tools toolbar above and circled in red, click on the Ruler icon which brings up the Measure tool.
As shown above, choose your units for your Distance and for your Area.
Choose the polygon icon in the Measure window and click the perimeter of the area that you are interested in measuring (shown in light gray above). You will get a linear measurement and an area, in units that you specify.
Finally, as all things ESRI, there was a little glitch when I tried this on two different data sets. I think you need to define the projection of the entire map for this to work. For instance, this technique worked beautifully with data in the above coordinate system (NY State Plane feet), but the area units were grayed out for a set of data in Geographic Coordinate System. If you figure it out, let me know because there's a world of people out there who are interested in the (easy) answer.