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I understand a radian is defined to be Arc/Radius, but why is it specifically defined this way? And how come this ratio works the way we expect angles to do?

In wikipedia it says: "Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc by its radius." But that seems to be the definition of "radians" not "angles"; is there a better definition?

EDIT: I understand radians have special properties (not that I already knew what these properties are).

I'm not interested in how special it is compared to other valid means of measureing angles, I'm interested in knowing how it is even considered valid a valid measurement of angles.

Also, what is an angle?

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Ratio sounds like a sensible definition, it is scale-invariant. However, an alternative is to decide that you are going to take a fixed circle of circumference $360$, and measure angles by the length of the circular arc on that specific circle determined by the angle. that gives us degree measure. Why $360$? Maybe it was because of the near coincidence with the length of the year, since angles were first systematically studied by astronomers. Maybe there is a connection between both and Babylonian base $60$ notation. – André Nicolas Feb 27 '12 at 19:19
Base 60 was used in Ptolemy's table of chords in the second century AD, and for probably more than a thousand years, that was the only trigonometric table extensive enough for practical use. – Michael Hardy Feb 27 '12 at 20:01
up vote 2 down vote accepted

If we define an angle to be a measure of rotation, with the initial side of the angle corresponding to the direction in which we initially face and the terminal side corresponding to the direction in which we face after the rotation, we must be able to calculate it without reference to the size of a circle (since we are considering rotation around a single point).

We know, geometrically, that the circumference of any circle is $2\pi r$ giving us the constant of $2\pi$ for what corresponds to a 'full rotation' when we divide out the (potentially misleading) size (radius) of the circle. Once we have defined the unit of measurement for rotation (based on a 'full rotation'), we then define larger and smaller rotations as fractions of a 'full rotation', i.e. as the same fraction of $2\pi$. This correspondence factors through the computation of arclength/radius, similarly giving us a constant measurement (regardless of the length of the arc) for the rotation corresponding to the angle through which we rotated.

To answer your question, radians are just a particular way of measuring an angle by measuring a length which is a fixed distance away from the point of rotation in such a way that the fixed distance does not affect the measurement of the rotation.

Going further, it is clear that angles cannot be measured by the amount of "space" between the initial and terminal sides, since the length those sides could provide more or less "space". Also, rotation cannot be measured with a ruler, since we are standing on the same spot and rotating - there has been no distance travelled. Thus we must project our rotation outward from the point of rotation (radius), in order to have a distance to measure. But, we do not want that radius to affect our measurement of rotation, and we can already see that projecting a radius of 2 feet gives a different distance travelled (arc length) than a projected radius of 1 foot gives for the same amount of rotation. These differences in arc length are solely based on the radius of projection, and when that radius is divided out, the ratios all end up equal for the same amount of rotation, regardless of the radius we use to measure. Thereby, we have a consistent method of measuring rotation.

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+1, A really lucid answer, thank you – seeker Jan 1 '14 at 18:31
"This correspondence factors through the computation of arclength/radius, similarly giving us a constant measurement (regardless of the length of the arc) for the rotation corresponding to the angle through which we rotated." I was reading this and im just a little confused as to what "correspondence" you refer to, is it (2pi)(r)=c, and more specifically what do you mean by "This correspondence factors through the computation of arclength/radius", i'd appreciate it if you could clear this up, thank you! – seeker Jan 4 '14 at 22:25
The correspondence I'm referring to is between the notion of a rotation other than 2$\pi$ and the measurement of an arc length corresponding to the same amount of rotation (on a circle of any size) after factoring out the radius of the circle used. For example, if we want to consider a "right angle", one-quarter of a full rotation, we would use $\frac{1}{4}$ of our $2\pi$, or $\frac{\pi}{2}$. If we measured out the intercepted arc length for this same angle on a circle of 10 units radius, we'd get an arc length of $5\pi$. Then factor out the 10 unit radius, voila, also $\frac{\pi}{2}$. – Andrew Parker Jan 7 '14 at 2:02
Thanks sir, that was a great help! – seeker Jan 8 '14 at 3:56

Radians are the natural units to use in calculus for the same reason $e$ is the natural base for exponential and logarithmic functions. The rate of change of the sign is a constant times the cosine. Only when radians are used is the constant equal to $1$.

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