Area/Perimeter Inquiry

For a shape of a given area, how many possible values of the perimeter might there be, and is there any relationship between the area and perimeter values? For example,

 

• A figure of area 1 square unit can only be of one shape and has a perimeter of 4 units.
• A figure of area 3 square units can take two shapes (are there any others?), with perimeter 8 units.