A tear drop can be represented by the parametric equations:
The arc length is:
Find such that . That is, how must I adjust the value of so that the arc length remains constant as I vary the parameter ?
Your definition of is independent of . Do you mean find such that ?
The way is defined it is proportional to so if you know A for the case r=1 it is simple to find A for any other value of r, or find the r corresponding to a desired value of A..