Surfer blanks the entire grid cell if the blanking BLN boundary passes through the cell. Thus the calculated area is often less than expected. Regrid the data with smaller grid cells (more grid lines) to improve the accuracy (or use Grids | Resize | Mosaic to resample the grid with an increased number of grid nodes).
In addition, be sure you are calculating the volume/area in the units that you think you are. The units of the area (and volume) results are X*Y(*Z), in the units of the grid file. If your grid file is in meters for X and Y, then the area results are in square meters. If the grid file is in lat/long, the area results are in lat*long, which is not very useful.
If you are comparing the area results to the area calculated by other means, also be sure you are comparing the same type of results (surface vs. planar area). Surface area results are generally larger than planar area results, as the surface area takes into account the area created by undulating surfaces.
Updated October 31, 2017