The best way I have found to do this is to 'create a cropped surface', which copies your surface to another file. I like doing it this way because it copies over your exact surface, without any other details (breaklines, points, etc.) that went into creating it. Often I'll get drawings from other companies and I don't want to have to reference their drawing and want to build my construction set to my liking.
Simply create the drawing you want to save the new surface to, then go to the 'surfaces' drop down menu > create cropped surface, and it walks you through the process from there. And you don't have to actually 'crop' the surface. I just create a boundary around the entire surface and select it that way, and any areas outside of the original surface are just omitted. Works pretty well.