The cheat was calculating tiny segments of a circle as the rope bent around the silo, multiplied by 2 for the other side of the silo. Then the obvious semicircle for the initial rope length area, minus the silo area.

Now if I can just figure out the exact area… :P

Diagram updated, same url. I still have no idea how to calculate an area from that though…

http://saladsammich.com/stuff/cow.svg

I don’t know the formula because I.. err… cheated (again). I figured that the cow has a semicircle of area in which he can move, which shrinks as he wraps himself around the silo. So I threw together a script to create an SVG image, which draws semicircles, and then told it to draw enough semicircles to create the illusion of a solid shape :)

So the cow has enough rope to go all the way around the circumference of the silo, right?

I’m not so great at creating formulas, but it looks to me that as the cow moves from the left side (anchor point) to the right side of the silo (opposite point to the anchor), the radius of his “freedom of movement” circle goes from pi to pi/2.

I would attempt to graph and/or formulamajize what that would look like but it’s late at night for me >.<

Plus, you’ll have an interesting answer every time someone asks why that fence is there! That’ll be what makes it worth it, right?

What if the silo was a polygon (with the same perimeter) instead of a circle? Here’s something you could try: Start with a line (area becomes a complete circle), then a triangle, square, pentagon, et cetera, and see what the area approaches as the number of sides approaches infinity. (I envy anyone with sufficient free time to actually try this approach, but at least I think it will deliver an answer… eventually. Maybe.)