###
An unusual and unintuitive solution

The solution shown here might seem a bit unusual since the trench is dug
partially outside the area enclosed by the circle. It may seem at
first that this could not possible yield a trench shorter than the circle
which is also sure to detect any straight pipe line. First, notice
that the length of this trench is about 5.14 km since the two straight
segments are are both equal to the radius, which is 1 km.
To see that the trench will detect any straight pipe line, notice that
if the pipe crosses the circular segment of the trench, then
it will be detected. On the
other hand, if it does not cross the circular trench segment,
then it must cross some part of the circle between the two dotted lines.
Since we can assume that it does not cross the circular part of the trench,
the pipe line is straight, it can not corss both dotted line segments. Being a straight line, it must, therefore, cross at least one of the
two straight trench segments.