Enclosing land by fence pieces You have one 44-meter piece of fence and 48 one-meter pieces of fence. Those fences are straight and cannot be bent. What is the biggest area you can enclose with those fences on a two dimensional plane?

It is obvious that the solution shape is a convex polygon, probably a strictly convex one.
Fix the 44-meter piece and consider the "loop" formed by the shorter pieces.
Take any three adjacent short pieces.
Keep their endpoints fixed and connect them with a line $L$.
This quadrilateral must have maximal area among those with the line $L$ as one edge and three other edges of length one.
In this "variational subproblem" two points (those between one-meter pieces) are free to move.
The solution to this subproblem is the symmetric one where the middle short piece is parallel to the long one.
(I do not have a rigorous proof right now, but this should be easy enough to check.)
This in particular implies that the two angles at the ends of the middle short piece are equal.
What we learn from the above is that any two adjacent angles between one-meter pieces must be equal.
Thus they are all equal.
Consequently all the 49 points lie one a circle.
It should be a straightforward calculation to find the area of this polygon.
