The area that you are seeing is calculated by Tatuk... not reported by the Trimble.   And it is an area calculated based upon "square degrees", which I'm sure you realize is a completely worthless number.   You will have to project the shapefile to a cartesian coordinate system if you want to acquire the area of the individual polygons.

Also, you will likely note that these are polygon strips, and not polygon sites.  The rest of the industry could deal with polygon sites.

Don's comments pretty well sum up the situation.   It appears that Trimble has chosen to believe that most of their customers don't have a complete crop recordkeeping system, and have chosen to provide summary data, rather that data that can be imported into a crop recordkeeping system. 

By the way... have you tried to view an elevation map from a 250 or 500, given that you might have high-resolution RTK elevation data being pumped into the unit?