Math Riddle: Find Optimal Amount of Fence for Farmer

Just a request that someone eventually post an answer (unless the answer is something ridiculous like blindfolding all the cows, in which case, I don't care).

There's an actual answer - not a trick question!

I can get it to 2.63896 down from 2.82843, assuming a unit square, but there may be a better trick.

Can you post the shape?

You can fence two adjacent sides, say north and west, and have another fence from the center to the southeast corner. That's 2+(1/sqrt(2))~2.707.

The number I got was from keeping the center to SE corner fence, and finding three straight fences to join the remaining three corners. I assumed these three meet at one point and that the point is on the NW-SE diagonal.

Are curved fences allowed?

Yes