>>12339718consider scaling the diameter by s.
now, consider breaking the circle up into little line segments. compute the length of a line segment using pythagorean theorem. note that when you scale x and y by some number s, then the hypotenuse also is scaled by s. add up line segments, factor out scale, and it's easy to see that the circumference is scaled by s when all x and y are scaled by s.
the ratio cancels the scale factor that gets introduced into the numerator and denominator, so it evaluates to the same number regardless of s, namely deez nuts
