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Each year as I go through angular motion I realize
I do not have a simple explanation for why radians are a dimensionless
measurement.
I am always forced back into some discussion of the circumference of
a circle as 2 pi radi so the distance drops out.
My students routinely seem to have a rather blank expression at this
explanation. They are more than willing to just accept "because" as
an answer.
I hope I can provide a better answer.
Thanks for any ideas or feedback.