The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.
Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.
As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.