Why do we not measure angles like this? An angle is a simple geometric figure that is obtained by taking a ray and rotating it about a point to form another ray, both of which have their starting point in common. The initial rays is called the initial side, the final one is called the terminal side and the common point is called the vertex.
The point of measuring angles is to describe, in a way to what degree (pun intended) the initial line is rotated about the vertex to obtain the terminal line. The three usual measures of angles i.e. degree, radian and grads are dimensionless since they are all based on measuring an angle as a ratio of the arc length to the radius (I might be wrong, please let me know if I am).
Now, let's say that we have an angle $AOB$, as shown here :

Here, $OP = OQ = 1~\mathrm{unit}$. $L$ is the length of the arc $PQ$.
So, why can't we call say that $L$ is the angle measure of $\angle AOB$ (in a new angle measurement system). For such a system, the measure of any given angle will be the same as it's measure in radians, only the dimensionality will be different. The dimensionality of angle measure in this system becomes the same as of lengths. I don't see how that's a problem, though.
Also, I find thinking about measuring angles like this much simpler than as a ratio of arc lengths.
So, can this be called a valid way of measuring angles. If not, why?
Thanks!
 A: The units on a measurement tell you, among other things, how the number will change if you decide to change the units you use.
Suppose that "$1$ unit" in your case is $1$ meter, and you measure the arc length $L$ to be $0.7$ meters. Reporting the angle as "$0.7$ meters" suggests that if we were using centimeters instead, the angle would be $70$ centimeters.
But that's not what actually happens. If we make "$1$ unit" be $1$ centimeter, then the arc will become $100$ times smaller, and the arc length $L$ will change to $0.7$ centimeters.
So it make sense to report the angle simply as "$0.7$" in both cases, because if we change the units we're using, the angle does not change.
A: Units of measurement are attached to values to represent their behavior under change of scale, or change of unit.
Consider a stick, whose length is measured in units of length. Let's say it is $x$ metres long. If we scale the object up by a factor of $c$, the stick will be $c\cdot x$ metres long. If we choose to measure in centimetres instead, the stick will be $100 \cdot x$ centimetres long.
Consider a carpet, whose area is measured in units of length squared. Let's say it has an area of $y$ square metres. If we scale the object up by a factor of $c$, the carpet will have an area of $c^2 \cdot y$ square metres. If we choose to measure in centimetres instead, the carpet will have an area of $100^2 \cdot y$ square centimetres.
In general, if a value in some system is measured in $m^n$, then scaling the system, or our unit of measurement, will cause that value to be scaled with the $n$-th power of the scale factor.
Now let's look at your angle example, and find the appropriate $n$ such that an angle should be measured in $m^n$. Let's say our initial unit of measurement is $m$ and the size of our angle is $1$ radian large. If we scale our angle up by a factor of $c$, it is still $1$ radian large, its value has multiplied by $c^0$. If we measure in $cm$ instead, the angle will still be $1$ radian large, it has been scaled by a factor of $100^0$. So it only makes sense for our angle to have units of length to the power of $0$, which means the angle is unitless.
