OK, here's a problem I'd like an analytical solution to:
A stupid farmer, prone to idiocy, buys a circular plot of land. Dunce. Anyway, he has one (count 'em - 1!!!) sheep that he feed by allowing it to graze on the field. Unfortunately, the farmer also like to graze on the field because he's too stupid to realize that (a) he can milk the sheep and get some solid good cheese; and (b) he can kill the sheep afterwards and make a nice mutton, cheese and grass sandwich on two pieces of hand. So the farmer is only willing to let the sheep graze on 1/2 of the field.
The farmer has one rope, which he might hang himself with if he can't solve this problem, but which he intends to tether the sheep with. Just 'cause he like to do things the hard way, the sheep will be tethered to the edge of the field, and a circular fence set up along the perimieter to keep the sheep from grazing on Farmer Sheisse's field across the way.
The question is, how long should the rope be? Report the answer as a ratio of rope length to field diameter.