Many fields are bordered by creeks, rivers, and lakes, thus irregular boundaries occur frequently.
The best practical information on measuring area is not found in math books, but in surveying books. You could xerox the map onto a piece of graph paper and count squares. If you had to do this type of thing very often, you would probably invest in an electronic or mechanical planimeter. Electronic planimetry Consider again measuring the area of a lake, given a good map. One could be a little more sophisticated by dividing the lake into geometric objects of known area, other than just squares.