But the angle is an adimensional unit (it's the ratio of two distances, one along the circumference and one along the radius) so 1 rad = 1. Therefore 1 degree is 0.0174... radians but it is also just 0.0174.
No, you're describing one particular way to measure angles. Radians express such a ratio, but degrees don't. 1° is not a ratio between distance along a circumference and radius, it's a ratio between amount rotated and complete revolution. 1° actually stands for 1/360 (of a revolution).
Which is why it's important to add the unit after the measurement. If someone tells you an angle measures 1, can you tell whether it's 1/360 of a revolution or the angle that would be formed by traveling along a circumference a distance equal to the radius of the circle?
Angles in the SI are a ratio of two lengths (and solid angles are a ratio of two surfaces), so degrees are also a ratio of two lengths. 1 degree is a ratio of pi/180=0.01745, which happens to be 1/360th of a revolution; and you have to write down the unit to indicate the multiplicative factor. But writing down radians is just for clarity.
