The optimal Area to Perimeter ratio is achieved by a circle.
Removal of any part of the circle's perimeter (circumference) to construct any other shape is counter productive.
Use the entire 36 meters for the circle (make your square
This minimization problem turns into a minimization on a closed interval. I haven't found a general post on the closed interval method. (Maybe the link owl will spot one.)
Then the sides of the square are each of length
The piece formed into a circle measures
For a circle,
The total area will be
Minimize the function of the interval:
The zero of
The critical numbers are
Calculate the Total area for the 3 values of