I think this is the real point of confusion. It is true, but far from obvious, that radians are the One True Way of measuring angles (for understanding the univ

I think this is the real point of confusion. It is true, but far from obvious, that radians are the One True Way of measuring angles (for understanding the universe, not necessarily for communicating to human beings). Once you believe in radians, the full angle is tau=2pi, so that is clearly the more fundamental number than pi. The reason why radians are the One True Way is the power series expansions for trig functions. Measuring the angle in radians makes e^ix = cos x + i * sin x work out where e^t = 1 + t + t^2/2 + t^3/3! + t^4/4! + ... (Exercise: derive the power series expansion for sin and cos from this) This formula connects geometry of circles to algebra and calculus in a fundamental way - there is no geometry needed to calculate e^ix, sin, or cos with power series (a computer could do it!), while cos and sin are all about angles and circles. Measuring angle in anything other than radians makes the formula horrendous (try it).