Why are radians more natural than any other angle unit? I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not).
I understand that using radians is useful in calculus involving trigonometric functions because there are no messy factors like $\pi/180$. I also understand that this is because $\sin(x) / x \rightarrow 1$ as $x \rightarrow 0$ when $x$ is in radians. But why does this mean radians are fundamentally more natural? What is mathematically wrong with these messy factors?
So maybe it's nice and clean to pick a unit which makes $\frac{d}{dx} \sin x = \cos x$. But why not choose to swap it around, by putting the 'nice and clean' bit at the unit of angle measurement itself? Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light $c = 1$). Sure, you would have messy factors of $2 \pi$ in calculus but what's wrong with this mathematically?
I think part of what I'm looking for is an explanation why the radius is the most important part of a circle. Could you not define another angle unit in a similar way to the radian, but with using the diameter instead of the radius?
Also, if radians are the fundamentally natural unit, does this mean that not only $\pi \,\textrm{rad} = 180 ^\circ$, but also $\pi = 180 ^\circ$, that is $1\,\textrm{rad}=1$?
 A: People call things "natural" when they simplify formulas.
Example, if there is a spinning wheel, the velocity $v$ of a point on the periphery is intuitively proportional to rotational speed $\omega$ and radius $r$.
If the rotational speed is measured in radians per second, then the exact formula and the intuitive one are identical:
$$v = r \omega$$
rather than something ugly like $r\omega(\pi/180)$.
A: 
I think part of what I'm looking for is an explanation why the radius
  is the most important part of a circle.

The most important part of a circle is the locus of points that comprise it.  Without that, you don't have a circle.
Radius is important in the definition of "circle" but the definition of "circle" is not identical with any circle.
The radian is defined as "the ratio between the length of an arc and its radius".
$\theta = s/r$
It is more "natural" than other angular measures for this reason:  the angle in radians is the normalized arc length, i.e., the radian measure of angle is the arc length for unit radius.  
EDIT: to address the numerous comments Zendmailer has made to other answers.
Zendmailer asks

What I'm asking now is, if they are indeed natural , how does the
  claim that 1 radian = 1 fit in?

For any angular measure $\alpha$, we have the almost trivial result:
1 $\alpha = 1$
So, the fact that 1 radian = 1 has nothing to do with the question of naturalness.
As I explained in a comment to another answer, the justification for the naturalness of the radian as an angular measure is geometric.
One can construct a circle with a length of string fixed at one end, the center of the circle, and a pencil.  Holding the string taunt, the pencil traces out the locus of points that comprise the circle.  The radius of the circle is the length of the string.
Having done that, what is the most natural way to measure length along the circle?  Lay the string along the circumference.  The arc length is precisely 1 radius.  The angle subtended by that arc length is a natural measure of angle, the radian.
The angle is the arc length divided by the radius so the radian measure of angle directly gives the arc length as a multiple of the radius.
A: Let me state some background facts which might be related to your questions and I hope they will help you understand the answers posted by others.


*

*There is a difference between units and dimensions. Every quantity that carries dimensions must carry units. 

*The opposite to the previous statement is not always true, for example angles have no dimensions at all because by definition they are length/length, but they have units. The unit in this case are used to identify the quantity as an angle. 

*Angles can be measured in degrees and can be measured in radians, just in the same way that distances can be measured in centimeters or in inches. Consequently there must be a conversion factor between the 2 units.

*Using $\pi$ radians = 180 degrees, you can see that $1 ~rad= 180^\circ/\pi=180^\circ/3.14\simeq 57.3^\circ$. That is to say 1 rad = 57.3 degrees (to put it in a form similar to something like 1 inch  = 2.54 cm).

*By definition $\displaystyle \theta=\frac{s}{r}$ rad, where $s$ is the length of the arc subtended the angle and $r$ is the radius of the circle. Note that the previous expression for the angle gives you the angle in radians. If you want it in degrees then it will look like this, $\displaystyle \theta = \frac{s}{r}\frac{180^\circ}{\pi}$. As you see the expression in radian is much simpler hence natural as pointed out by Mike Dunlavey. 

*If you have a particle that is rotating around a circle of constant radius $r$, then from the equation $\displaystyle \theta=\frac{s}{r}$ rad you can see that we can get $\displaystyle \omega=\frac{v}{r}$ rad per unit time (where, by definition, $\omega = \frac{d\theta}{dt}$ and $v=\frac{ds}{dt}$). Again, as pointed out by Mike, the equation for the angular velocity will have an extra factor of $180^\circ/\pi$ had we wanted the angular velocity be expressed in degrees per unit time instead of rad per unit time.

*When an angle, expressed in radians or degrees, multiplies a unit of distance say, the surviving unit is that of the distance. For example: given $\omega = 2 rad/s$ and $r=1 cm$, hence $r\omega = 2 cm/s$. This is why in this case you can say 1 rad =1.

A: Consider the Taylor series for the trigonometric function. For instance sine
$$ \sin \alpha = \alpha - \frac{\alpha^3}{3!} + \dots = \sum_{n=0}^\infty (-1)^{n}\frac{\alpha^{2n+1}}{(2n+1)!},$$
or cosine
$$ \cos \alpha = 1 - \frac{\alpha^2}{2!} + \dots =\sum_{n=0}^\infty (-1)^n \frac{\alpha^{2n}}{(2n)!}.$$
If you were to choose some other unit for angle these very tidy series would pick up some additional factors in every term. 
That kind of thing is "unnatural" to mathematicians.
A: Most importantly
$$ e^{i x} = \cos x + i \sin x$$ only holds (in this form) in radians.
So now you might ask why $e$ is more natural than any other number ;-)
A: Angles are defined as the ratio of arc-length to radius multiplied by some constant $k$ which equals one in the case of radians, $360/2\pi$ for degrees. What you're effectively asking is what's natural about setting $k$ = 1? Again it's tidyness as pointed out in dmckee's alternative answer.
A: The reason radian was adopted was that it was easy to relate with the circumference of a circle as 2*Pi if the radius was one unit. There is no such thing as 360 degree(it was a misconception in early times that one year is made up of 360 days so they took it 360). From the present day statistics it shall be 365 1/4 but it doesn't change calculations and results gets adjusted automatically on calculation.
Calculations were easy to manipulate with Pi rather than Degree,minutes,second and they are both interchangeable.
So, a comfort became a tradition.
