There's a much simpler way to say this, I think. It is already the case that we relate everything to the unit radius---hence "radian". (If we tried to relate

There's a much simpler way to say this, I think. It is already the case that we relate everything to the unit radius---hence "radian". (If we tried to relate everything to the diameter instead, a lot of things would break much more deeply, starting with the very convenient derivative of sin(x).) Given that, ask yourself this: do we want a measure of the semicircle, or of the circle? The former is \pi. The latter is \tau.