Why are angles in "degrees" dimensionless?

Question

Angles measured in radians are considered to be dimensionless because the radian measure of angles is defined as the ratio of two lengths $\theta=\frac{s}{r}$ (where $s$ is some arc measuring $s$-units in length, and $r$ is the radius). However the degree measure is not defined in this way and it is said to be dimensionless too.

Why are angles in degrees or angles in general said to be dimensionless? Angles measure an "extent" of something, in this case rotation, so why are they dimensionless? What is meant in mathematics when something is said to have no dimensions?

Answer 1

Good question, towards the end. I'll try to answer it but first I'm going to take issue with the beginning of your question.

I don't think it's correct to say that "radians are dimensionless". It is a measurable quantity that has dimension (or no dimension) and not a unit of measurement. So your question as to why angles are dimensionless is a good one, but the distinction between radians and degrees is just a distraction.

To see that there is no difference (for this discussion) between radians and degrees, consider the respective arc length formulae, $$s=r\theta\quad\hbox{and}\quad s=\frac{2\pi r\theta}{360}\ .$$ These might look significantly different, but that's only because the first formula contains an invisible item. Making it visible, $$s=\frac{2\pi r\theta}{2\pi}\quad\hbox{and}\quad s=\frac{2\pi r\theta}{360}\ .$$ The only difference is in the constant, and that makes perfect sense because radians and degrees are proportional to each other. It's just like formulae for area of a square in terms of a side: if you measured the side in metres and the area alternatively in square metres or square feet (the latter, admittedly, crazy) you would get two formulae which would be identical except for the value of a constant. Not only that, but in the first case the constant would be "invisible", as in the above example.

BTW I know you said not to mention radians, but I hope you don't mind that I have mentioned them purely in order to explain why we don't need to mention them ;-)

So, to the main question: why are angles dimensionless? IMHO the answer is that this is a matter of convenience and not of necessity. Suppose that we assign lengths the dimension $L$ as usual, and angles the dimension $A$. Our arc length formula in terms of degrees would be $s=cr\theta$, where the constant $$c=\frac{2\pi}{360}$$ has dimension $A^{-1}$. In terms of radians, again $s=cr\theta$, where in this case the "invisible" constant $$c=1$$ has dimension $A^{-1}$. Note that there is no problem with a constant having dimension. Consider two familiar force formulae: $$F=ma\quad\hbox{and}\quad F=\frac{Gm_1m_2}{r^2}\ .$$ If we take the constant $1$ in the first to be dimensionless, then force has dimensions $MLT^{-2}$, and so the gravitational constant $G$ has dimensions $M^{-1}L^3T^{-2}$: it isn't dimensionless. If on the other hand we insisted on $G$ being a dimensionless constant, then the constant $1$ in $F=ma$ would not be dimensionless.

Answer 2

Imagine you are a surveyor and you have to map a building. You need meters and goniometers for that. Of course, you are not going to use the same tools to measure the perimeter of the parking lot and the size of the tiles in the kitchen floor. For the second one you'll need a smaller meter, and also a smaller unit of measure of length. You need to record that the side of the parking lot measures 10 meters, while the side of a tile measures 10 centimeters: the number 10 alone means nothing, when associated to a length.

On the other hand, even if you are probably going to use a smaller goniometer to measure the angle of the kitchen tile, there will be no need to take note of that. Indeed, the goniometer just tells you how much of a circumference is covered by the angle you are measuring. A right angle covers one quarter of a circumference, and the number $1/4$ alone is a perfectly good measure of an angle. That's what "dimensionless" means. "Radians", "degrees", "centi-degrees" are just different ways of writing fractions.

Answer 3

Arc length is measured in distance. Radius is measured in distance. Angle of arc is arc length over radius, thus the units cancel out, hence Radians is ephermeral, or dimensionless.

I like to pretend Radians is an actual unit, for the purposes of verifying that my units are correct. Same with Steradians.

Answer 4

Degrees can be defined in terms of a (dimensionless) constant times a radian, so naturally, if radians are dimensionless, then degrees must be too. That something has no dimension simply means it's a scalar. If it helps, just look at the dimension as a variable as well, in this case $m$ for meters:

$\frac{a \cdot m}{b \cdot m} = \frac{a}{b}$

The dimensionalities get cancelled out. A ratio of two quantities with equal dimensions must naturally be dimensionless.

Also, $\sin(x)$ and similar functions aren't defined for anything but actual NUMBERS. You can't take the sine of an angle if the angle isn't a number and nothing but a number, that's simply not defined.

Answer 5

To define some dimensionless ratios of measurable physical quantities by means of a comparative multiplier or fractional number.

Angles are dimensionless not only in degrees, but also in radians. It is a ratio of two lengths. The ratio is between a circular arc length and straight length of circle radius.

Lengths in meters are compared to a standard bar in kept in Paris.

Coefficient of friction is ratio of two perpendicular forces.

Mach number is a dimensionless ratio number comparing the speed of anything to that of sound.

Sound intensity itself is measured in decibels, multiplied by 10 for convenience for two power ratios.

Answer 6

To your question “why are angles in degrees or angles in general said to be dimensionless?” the short answer is that regardless of the unit used to measure angles, the measurements include ratios of quantities of the same kind, length ratios (for example $\,\mathrm{m}/\mathrm{m})$), and are therefore quantities without units, quantities of dimension one, i.e., they are dimensionless.

To your question, “What is meant in mathematics when something is said to have no dimensions?” the short answer is that, in this case, it refers to a physical quantity that is a number and has no unit of measurement.

However, we consider it appropriate to provide detailed answers.

There are several units for measuring angles: degrees, gradians, and radians. The procedure is similar for all units of measurement: We place the vertex of the angle at the center of a circle, measure the length of the arc intercepted by the rays of the angle, denoted by $\,s$, an compare it with the length of the arc taken as a unit, denoted by $\,s_u$.

We can answer these questions in two different ways.

First answer:

If we accept that degrees, gradians, and radians are dimensionless units, then the measure of the plane angle $\,\theta\,$ is equal to the ratio between $\,s\,$ and $\,s_u\,$ $$\theta = \frac {s} {s_u}$$

To show that the formula works correctly:

1.- When angles are measured in degrees, the circumference $\,C\,$ is divided into $\,360\,$ parts of equal length, therefore $\,s_u=C/360$. An angle of $\,1\,$ degree, $\,1°$, is one whose rays pass through the ends of one of those parts. Given that for a complete turn $\,s=C$, in this case the angle measures $$\theta=\frac {C} {C/360}\qquad \implies \qquad \theta=360° $$ 2.- When angles are measured in gradians, the circumference $\,C\,$ is divided into $\,400\,$ parts of equal length, therefore $\,s_u=C/400$. An angle of $\,1\,$ gradian, $\,1^g$, is one whose rays pass through the ends of one of those parts. Given that for a complete turn $\,s=C$, in this case the angle measures $$\theta=\frac {C} {C/400}\qquad \implies \qquad \theta=400^g $$ 3.- When angles are measured in radians, we know from the SI Brochure that “One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius” (p.138), therefore $\,s_u=r$, where $\,r\,$ denotes the length of the radius. Since for a complete turn $\,s=C=2 \pi r$, in this case the angle measures $$\theta=\frac {2 \pi r} {r}\qquad \implies \qquad \theta=2 \pi\, \mathrm{rad} $$ The unit for measuring angles in the International System of Units (SI) is the radian, which is considered a dimensionless derived unit and $\,1\,\mathrm{rad}=1\,$.

Second answer:

If degrees, gradians, and radians are not dimensionless units, then in $$\theta = \frac {s} {s_u}$$ the variable $\,\theta\,$ must be dimensionless, $\,\dim⁡ \theta=1\,$ since $\,s/s_u\,$ is dimensionless because the units of length cancel out (e.g., $\,\mathrm{m}/\mathrm{m}$). The difference is that $\,\theta\,$ has no units, but is simply a number. In these cases, let us denote $\,\beta\,$ as the value of the angle.

To show that the formula works correctly:

1.- When angles are measured in degrees, $\,s_u=C/360$. For a complete turn, $\,s=C$, in which case $\,\theta\,$ is the “number of degrees” and the angle $\,\beta\,$ measures $$\theta=\frac {C} {C/360}=360\qquad \implies \qquad \beta=360° $$ 2.- When angles are measured in gradians $\,s_u=C/400$. As for a complete turn, $\,s=C$, in this case $\,\theta\,$ is the “number of gradians” and the angle $\,\beta\,$ measures $$\theta=\frac {C} {C/400}=400 \qquad \implies \qquad \beta=400^g $$ 3.- When angles are measured in radians, $\,s_u=r$, where $\,r\,$ denotes the length of the radius. For a complete turn, $\,s=C=2 \pi r$, in which case $\,\theta\,$ is the “number of radians” and the angle $\,\beta\,$ measures $$\theta=\frac {2 \pi r} {r}=2 \pi \qquad \implies \qquad \beta=2 \pi \,\mathrm{rad} $$ The difference between $\,\theta$, which represents numbers, and $\,\beta$, which represents angle measures that have a numerical value and a unit of measurement, is clear.

For many years, the status of the radian in the SI Brochure has been the subject of debate in the Consultative Committee for Units (CCU) of the BIPM. There is still no consensus. The majority believes that the radian is a dimensionless derived unit (and therefore remains so in the SI Brochure), while a group considers that the radian should be a base unit, as it is not a dimensionless derived unit.

We believe that the correct answer is the second one.