Four farm-hands need to carefully walk the entire perimeter of a large square field to check for signs of infestation. They can each walk separately and any section of edge need only be checked by one person. The field is 200m on an edge and they all start at the same corner. Each person can either walk normally at 2m/sec or walk while checking at only 1m/sec. Any person may cut through part of the field at a normal walking pace. They must all finish at the opposite corner of the field.

What is the shortest time in which they can check the entire edge?

Consider the same problem with a circular field of radius 100m.

How long would this take?

Charlie's solution on the circle looks reasonable to me, although I haven't checked the math.

However, there is a faster strategy on a square.

If they all start at the NorthWest corner, then one farmhand can
inspect more than half the north edge, cut the corner at a 45 degree
angle, and inspect more than half the east edge. Another farmhand
can walk the north and east edges, inspecting only the corner which the
first farmhand does not walk. This will be faster than having
everybody walk two edges.

Sorry, I don't have time to do the math. I'll do it later, if no one else does it first.