What is the largest triangle (area) that can be fitted into an area
bounded by the boom, mast (between gooseneck and backstay crane),
backstay crane and backstay?

This is basically the problem that is causing such a kafuffle in a few
other threads here.

Well actually it's not going to be a triangle because the boom does not
extend to the backstay. We can increase the area bit by bowing out the
leech of the sail to just reach the backstay at a point roughly
perpendicular from the position of the clew on the boom, this is roach.

For structural reasons the top of the headboard is not pointed but
rather chopped of parallel to the foot and this is partly why the
backstay is carried aft of the mast head by the crane, allowing the head
board to hoist a bit higher.

Now hear is the rub: If we lower the headboard by a foot and increase
the curvature of the roach to meet the headboard at the new lower
position we REDUCE the area of the main. The roach, expressed as a
percentage of the area of the main increases. This happens for two
reasons: 1) The leech is now at a position further forward of the old
on, so the distance from the point of maximum roach to the line
connecting the clew and head is a bit larger. 2) The area of the
triangle defined by luff, foot, and line connecting clew to head is now
smaller. Consequently the roach percentage rises.

BUT, the overall area of the sail is LESS!

Now why would you not want a sail that hoists to the black band?

Cheers
Marty