A local farmer wants to keep his sheep safe. He has one............what shape will the fenced off area be and how large?

Question

A local farmer wants to keep his sheep safe. He has one mile of fence, with which to surround sheep. Luckily there is a very straight river which he can also use. Obviously the farmer wants to fence off as much land as possible, what shape will the fenced off area be and how large?

Answer 1

Answer:
The fenced of area will be a large semicircle, with the river forming the diameter of the circle and the area will be 0.16 of a square mile.

As the fence forms the curved semicircle section, so half the circle has a length around of 1 mile (the fence), so the whole circle has a circumference of 2 miles. As circumference = Pi x diameter, then diameter = circumference / Pi. In this case the diameter is therefore 2 / Pi and the radius is therefore 1 / Pi.

The area of the entire circle would be Pi x (1 / Pi) x (1 / Pi) = 1 / Pi.

We only want half of this area, so the answer is 1 / (2 x Pi) = 0.16 of a square mile.

Note: if the farmer were to make a square area, each side would be 0.25, and 0.25 x 0.25 = 0.0625 of a square mile, which is smaller.