Confusion with properties of radians I have a problem that goes like this:
"In a circle with radius $6$, what is the measure (in degrees) of an arc whose length is $2\pi$?"
I thought that the answer would be $360$ degrees because $2\pi$ radians is one circle, but the answer in the book is $60$ degrees. The book solved it like this:
$$Circumference=2\pi r=12\pi$$
$2\pi$ is $1/6$ of the circumference:
$$1/6\times360=60$$
This route makes sense to me, but I am really confused on the part that it said the length of the arc was $2\pi$, and $2\pi$ radians is $360$ degrees. I drew the conclusion that it meant radians because I was told that you assume that a measure is in radians unless you are told that it is something else. I have just learned about radians, and I obviously made a mistake. Can someone explain to me where I went wrong?
 A: By length they mean the arc length, not the angle measure. 
A: Arc arc of $2\pi$ rd has length $2\pi$ on a circle of radius $1$. The length of an arc of $\theta$ rd is equal to $\theta\times\text{radius}$. Whence, if the length is $2\pi$ the arc measure in radians is $\dfrac{2\pi}6=\dfrac\pi3$, i.e. $60$°.
A: The length of an arc is equal to $r\theta$ where $r$ is the radius and $\theta$ the measuer of the angle in radians so we have $$r\theta= 2\pi$$ Hence, since $2\pi=360^{\circ}$ it is obvious that the asked angle is equal to $\frac{360}{6}$.
Thus the answer is $60^{\circ}$  
A: First of all, a radius is a length (not a number).  Similarly, an arc (length) is a length (not a number).  We have to assume that, in the stated problem, the radius and the arc-length are both measured in the same length units, say LU.
Secondly, the (dimensionally correct) relationship between arc-length, s, radius, r, and central angle, theta, is:
s = r theta/rad
[not "s = r theta"--as is commonly (almost universally!) written.]
where rad is a reference angle of one radian.  [The quantity theta/rad is sometimes called the "radian measure of the central angle"--but unfortunately (and confusingly) written as "theta."]
In this case we have: s = 2 pi LU, r = 6 LU.  So:
2 pi LU = (6 LU) theta/rad
giving: theta = (2 pi)/6 rad = (2 pi rad)/6.  Since 2 pi rad = 360 degrees (= 1 rev), the answer is:
theta = 60 degrees.
--B P Leonard, Emeritus Professor of Mechanical Engineering, The University of Akron.
