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In $s=r\theta$, is $\theta$ unitless? Isn't it measured in radians? If angles are unitless, what are radians and degrees? Please clarify.

Matti P.
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Niraj Raut
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  • If you read your source material carefully, I'm sure they mention that $\theta$ should be in radians. The whole point of radians is that formula. – Matti P. Sep 07 '20 at 11:46
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    Angles are dimensionless, not unit less. – lulu Sep 07 '20 at 11:46
  • @lulu Is radians a unit? – Niraj Raut Sep 07 '20 at 12:40
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    Yes. And degrees are also a unit. – lulu Sep 07 '20 at 12:40
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    @lulu What is meant by "dimensionless"? – Niraj Raut Sep 07 '20 at 12:41
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    They don't scale with length. If you form an angle with two line segments that share an endpoint, and then you scale up all the lengths in your field, the angle is unchanged. – lulu Sep 07 '20 at 12:47
  • This question has appeared before on this site, here for example. And, of course, it has appeared on the Physics site, here for example. – lulu Sep 07 '20 at 12:51
  • @lulu: the idea is to measure an angle as the length of the corresponding arc of the circle of radius 1 If you rescale the length, the arc will also change and its length will be the same. – Andrea Mori Sep 07 '20 at 12:55
  • there is no physical dimension assigned to angles. they have dimension 1, which is counter intuitive, but in SI units, it is identical to the SI unit 1. it’s just that these dimensionless units still give meaning to things. for instance, rotational frequency. if something is rotating and I ask you the for the rotational frequency and you tell me 50 per second, then neither of us will know what the frequency is. you must give me a unit, for example radians, to tell me the frequency is 50 radians per second. – C Squared Sep 07 '20 at 12:58
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    @AndreaMori It's the same whether you speak of angles as geometric quantities (as I did) or you speak of a way to quantify angles (as you are doing). Your way makes it appear as if they should have the dimension of length, but they don't...either you require that the radius be $1$ (as you did) or you divide by the radius. Either way, it is a scale invariant. – lulu Sep 07 '20 at 12:58
  • @lulu So radians are dimensionless but they are units, right? – Niraj Raut Sep 13 '20 at 10:40
  • You asked me that I already and I confirmed that they are units. – lulu Sep 13 '20 at 11:37
  • @lulu Aren't radians unitless as it is the ratio? – Niraj Raut Sep 13 '20 at 14:04
  • You are asking the same question over and over. The ratio you speak of is a way to measure the angle, the angle itself exists independent of the way you measure it. If you use arclength on the unit circle you get radian measure. If you declare that the circle is $360$ and measure the subtended portion, you get degrees. Nothing special about $360$, obviously. I could declare the full circle to be $1517.981$ and I'd get a whole new set of units with which to measure angles. all of these systems of units are fundamentally the same, they just differ by a multiplcative factor. – lulu Sep 13 '20 at 14:20

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Angles are usually measured in degrees, or radians.

By convention the angle corresponding to the full circle is divided in 360 equal parts, of degrees. The convention goes back to ancient times and 360 is convenient because it has a lot of divisors. Thus it is measure that it is useful for practical purposes.

Mathematicians do measure angles in radians. By definition the radians measure of an angle is the length of the radius $1$ arc defined by the angle itself.

Thus, the full circle is an angle of $2\pi$ radians as that is the circumference of a circle of radius $1$. Mathematicians prefer radians because the formulae

$$ \frac d{dx}\sin(x)=\cos(x),\qquad \frac d{dx}\cos(x)=-\sin(x) $$ hold only when $x$ is measured in radians.

Andrea Mori
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