TITLE: Calculating the area of an irregular polygon
QUESTION [12 upvotes]: Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? 
Thanks in advance

REPLY [2 votes]: Following on @aelguindy's comment, this was an area of research by one Prof. David Robbins (who passed away prematurely).  Here is a typical paper from that research effort: he was able to find explicit expressions for the areas of cyclic heptagons and octagons.  Anyone who browses through that paper will understand that this problem is extraordinarily difficult in general.
For amusement, I compared relatively simple results I got from a Putnam exam problem with his result and got agreement, happily.  Here is the problem:

Find the area of a convex octagon inscribed in a circle that has 4
  consecutive sides of length 3 units and 4 consecutive sides of length
  2 units. Give your answer in the form $r+s \sqrt{t}$, where $r$, $s$
  and $t$ are positive integers.

I'll leave it as an exercise for the reader.