strenght of relationship and nonlinearity of r

If one goes from r (correlation coefficient) 0.2 to 0.3 (0.1 difference) or from 0.8 to 0.9 (same difference) the strenght of the relationship is not the same; in other words the strengt of the relationship between variables is not a linear function of r. How do we know this? Which test is used to determine strenght of relationship so we can demonstrate that and why is that method not used?