There is no greatest perimeter. The longer and skinnier you make the rectangle,
the greater its perimeter becomes, even though its area remains the same.
Look at this:
√39 by √39, Area = 39, Perimeter = 4 √39 = about 24.98
6 by 6.5, Area = 39, Perimeter = 25
3 by 13, Area = 39, Perimeter = 32
2 by 19.5, Area = 39, Perimeter = 43
1 by 39, Area = 39, Perimeter = 80
0.1 by 390, Area = 39, Perimeter = 780.2
0.00001 by 3,900,000, Area = 39, Perimeter = 7,800,000.2