# Why is an angle dimensionless?

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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The answer to both your questions is the same - trig ratios and radian measure are both dimensionless because they are defined as the ratio of two lengths, which have the same units so they cancel. – Ragib Zaman Jan 30 '12 at 9:57
@RagibZaman In that case does it mean that Dimensional Analysis cannot be applied to equations which involve ratios of 2 quantities with the same unit? – Green Noob Jan 30 '12 at 10:03
Further, how can we extend this logic for angles in degrees? I don't think it is defined as a ratio of two lengths. – Green Noob Jan 30 '12 at 10:08
@GreenNoob: No it does not mean Dimensional Analysis cannot be applied, it just means such ratios have an empty dimension. If they are equated or compared to an expression with a non-empty dimension, then there is an error, but if they are equated or compared to another such ratio or an explicit number, then no error is detected. – Marc van Leeuwen Jan 30 '12 at 10:09
When doing dimensional analysis on a problem which has an angle as a parameter, you generally find that the solution can involve an arbitrary function of the angle, as in e.g. the problem of how far a ball travels under a gravitational field $g$ if thrown with velocity $v$ at angle $\theta$ (the dimensional analysis solution is $x\propto v^2/g \times f(\theta)$ – Chris Taylor Jan 30 '12 at 10:34