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The angle is adimensional. This is clearer when you measure angles in radians: in this case, the angle is part of a circumference of a circle, divided by the radius. Since you are dividing two lengths to obtain this, any dimensions disappear.
Since radians are adimensional, any other measurement for angle must also be adimensional. For example, to convert radians to degrees, simply multiply by (180/pi), which is a "pure number" (no physical measurement, no dimensions).
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