Wriggling on a hook

A worm of unit length is caught on a hook. He can only move in the plane i.e not in three dimensions. He can not exert any momentum against the hook which means that his center of gravity always will be on the vertical line from the hanging point to the ground. Suppose now he wriggles around as much as he can.

What is the area covered by the worm during his wriggling?

A value for the area and a short description of how you found it will suffice as an answer. Solve this puzzle and you will enter the "Fields of Gold" hall of fame together with Pooh (who solved the first cornfield puzzle and the fuse burning puzzle) and future master puzzle solvers.