Imagine a circle and a central angle in it. If the length of an arc that this angle cuts off the circle equals to its radius, then, by definition, this angle's measure is 1 radian. If an angle is twice as big, the arc it cuts off the circle will be twice as long and the measure of this angle will be 2 radians. So, the ratio between an arc and a radius is a measure of a central angle in radians.
For this definition of the angle's measure in radians to be logically correct, it must be independent of a circle.
Indeed, if we increase the radius while leaving the central angle the same, the bigger arc that our angle cuts from a bigger circle will still be in the same proportion to a bigger radius because of similarity, and our measure of an angle will be the same and independent of a circle.
Since the length of a circumference of a circle equals to its radius multiplied by #2pi#, the full angle of #360^0# equals to #2pi# radians.
From this we can derive other equivalencies between degrees and radians: