Intuitively the formula should have the length of the rope, radius of the fence, and PI. And intuitively a rope length of the radius would cover half the circle

Intuitively the formula should have the length of the rope, radius of the fence, and PI. And intuitively a rope length of the radius would cover half the circle, but a quick test of putting two goats in there shows they can't eat all the area. So the formula probably have a minus in it. Next I would try to figure out how large the goat circle outside the fence is because then i would also know the area inside...