I thought I (once, long ago) understood that the official area for racing purposes was defined as the triangle - which was one of the reasons that spinnakers were such popular racing sails. I suppose you could produce a decent approximation of true (cloth) area by treating it as a quarter-sphere of radius J (or whatever is your pole length) plus a half-cylinder of the same radius, height measured or estimated, to allow for the lower part of the sail.