Radian describes as the length of the arc divided by the radius of the arc, θ = s / r (Wikipedia)
its very cleary that θ is not dimensional because its the ratio between two numbers of the same unit( arc length to radius length), but then why θ has a unit(radian)?
in digital communication, we have SNR(signal power to noise power) the SNR is unitless, why θ is different?
Why is the Radian not unit-less
definitiongeometry
Best Answer
Just because a quantity is dimensionless, doesn't mean it's not convenient to give it a "unit". It would be weird to call a subtended angle "$0.26$" when it's $0.26$ radians, if only because we're used to every quantity telling you what it's $0.26$ of. You certainly wouldn't want to confuse radians with the steradians of solid angle.
Angles aren't alone in this. Toxicity is often measured in LD50, the amount needed to kill 50% of specimens. But it's measured relative to body mass, e.g. as 3 mg/kg as opposed to just saying $3\times 10^{-6}$.