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I was trying out Dimensional Analysis on a few equations and realized that angles have no dimension. Otherwise equations such as $s=r\theta$ are not dimensionally consistent.

Further, why don't trigonometric ratios have any dimension?

PS: I couldn't find any appropriate tag for this question. Could someone re tag as appropriate? Thanks.

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    It's possible to treat angle with dimensions, but then you would have to recast the dimensional analysis of all things, and some quantities could be split as a result. In Leo Young's book on EM units, solid angle appears as a specific dimension. The definition of a radian as $\frac{metre}{metre}$ supposes that the constant of circumference to radius is dimensionless, whereas in $C = k R \theta$, $k$ would have units of $degree^{-1}$. A similar dimensioned constant appears in the CODATA (1000 mol/kg), where the previous was purely a numeric ratio.2013-07-14

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Compare to a ratio of weights, it is weights that you compare. You get no units for the ratio, right ? I mean the result is independent of choice of units. But with angles your ratio is with lengths ! not angles ! so don't be surprized that you get "units". Any partition of an angle is named with "units" The funny thing is that unit transformation with angles obeys the same laws as with any other unit.