Here is a small AutoLISP program I did a while ago. It's freeware so just grab it, use it, change it or contact me if you want something else to be done.
It makes use of the FIELD functionality in AutoCAD but makes this specific operation easy and quick.
Run the command AT and select a polyline and then where to place the text.
The result is in SQ. FT. but the AutoLISP code can easily be changed for other formats.
When the polyline is changed the area is updated