Be cause the ratio circumference: diameter is the same for every circle, is the ratio circumference: radius the same for every circle?

Absolutely !  You have just unlocked one of the great
immutable secrets of the universe.

Ratio of
               [ circumference / diameter ] =
ratio of
               [ circumference / (2 x radius) ] =
(1/2) ratio of
               [ circumference / diameter ] .

I don't know how else to say it, so I hope that did it for you.