Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:

theta = arc len / r


Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?

In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.

How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?

• Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant. – leonbloy May 21 '14 at 11:47
• A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe. – MJD May 21 '14 at 13:07
• @MJD It makes sense. But I can tell the same for degree, isn't it? – Viacheslav Kondratiuk May 21 '14 at 13:43
• Degrees are also dimensionless. – MJD May 21 '14 at 13:46

A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.

However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.

• and what about degrees? are they also dimensionless? – Viacheslav Kondratiuk May 21 '14 at 11:22
• $1$ degree is $\frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else. – Graham Kemp May 21 '14 at 11:31

The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.

For example, you quote the arc length as $s = R\theta$. This is obviously no longer homogeneous in dimensions if $\theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $\theta$, and that $s=R$ for $\theta=1\,\mathrm{rad}$, we conclude that the true form of that relation must be $$s = R\frac{\theta}{1\,\mathrm{rad}}.$$ (When asuuming that $\mathrm{1\,rad = 1}$ is dimensionless, it yields the original expression.)

Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$\exp(x) = \sum_{k=0}^\infty\frac{x^k}{k!}.$$

If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $\sin(\theta/\mathrm{rad})$ when talking about the sine of an angle.

As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.

What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $\omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $\omega$ would be $\mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.

If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.

• Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $\omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2\pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse. – Evgeni Sergeev Jul 26 '17 at 14:57
• Why we need to refer the $rad$ if it is actually dimensionless? – user599310 May 28 '20 at 21:18
• Mathematically speaking, the rad unit is redundant (if you define it to be dimensionless). But semantically, there is huge conceptual difference between e.g. frequency and angular velocity. Writing the 'rad' explicitely adds clarity and helps express the author's intend. It again is a matter of convention. – polwel May 29 '20 at 7:38

Yep, you can work with dimensioned angles.

The formula for the arc length is

$$l=\aleph\theta,$$ and that for the area of a sector

$$a=\frac{\aleph\theta r^2}2.$$

The universal constant $$\aleph$$ is in per-angle-unit, and $$\aleph=1 \text{ rad}^{-1}=0.0174533\cdots\text{ deg}^{-1}$$.

For example, when considering an harmonic movement,

$$e=A\sin(\aleph\omega t)$$ where $$\omega$$ is in angle-unit $$s^{-1}$$ and $$e$$ in $$m$$.

$$v=\dot e=A\aleph\omega \cos(\aleph\omega t)$$ is in $$ms^{-1}$$.

Another universal constant worth to know:

$$\Pi=3.141593\cdots\text{ rad}=180\text{ deg}.$$

denotes the aperture angle of a half-circle.

It fufills

$$\aleph\Pi=\pi.$$

Hence the famous Euler formula,

$$e^{i\aleph\Pi}=-1.$$

A quantity is dimensionless if it has same magnitudes in different units.

$$1 \text{rad}=\dfrac{1m}{1 m}=\dfrac{1 nm}{1 nm}=\dfrac{1\text{light year}}{1\text{light year}}=1$$ As you noted The same units get cancelled.

However length is not so.

$$1m=100cm$$

$1\ne 100$ for obvious reasons.

Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.

As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.

A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.