The curved path edges are in a form of two pairs of quadrants. One pair is shown below:
We can find the lengths of the inner and outer arcs using the radii of 4m and 6m respectively.
The formula to find the length of the arc of a quadrant is

Length of inner arc:

Length of outer arc:

Hence, the length of the pair of arcs is

For the two pairs, we have the length to be

Hence, it will take approximately 32 meters of plastic edging to complete the project.