The fenced off 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 (using the river as one side), each side would be 1/3 of a mile, and 1/3 x 1/3 = 0.11 of a square mile, which is smaller.