Angle measurement - radian and degree

I want to present my reasoning here about angle measurement and to get some opinions about its correctness and usage.

One radian denotes the angle that has the center of the circle as a starting vertex and two rays that cut off an arc $l$ with length equal to the circle's radius.

In fact, how many radians denotes how many arcs with length $r$ the angle cuts off. For example, the circumference of the circle is equal to: $2\pi r$. To find how many arcs with length $r$ are there or with other words, how many radians, we use this approach: $$\text{how many arcs} = \frac{\text{circumference}}{\text{radius}} \tag 1$$ If we denote how many arcs as $\theta$ we have: $$\require{cancel}\theta = \frac{2\pi \cancel{r}}{\cancel{r}} \tag 2$$ And we get: $$\theta = 2 \pi \text{ arcs}(\text{rad}) \tag 3$$ This means that the angle whose arc has length $2 \pi r$ has $2\pi$ arcs with length $r$. Now, if we have the length of the arc $l$ we can calculate how many arcs of length $r$ an angle has or with other words it's size in radians. Here is the formula: $$\theta = \frac{l}{r} \tag 4$$

One degree is $\frac{1}{360}$ of the whole circle. $$1^\circ = \frac{1}{360} \tag 5$$ This means that the circle has $360^{\circ}$ or $360\times\frac{1}{360}$ parts (pieces).

Mixing radians and degrees:

Since the circle has $2\pi$ arcs(radians) and we can also divide it on 360 pieces we can write the following equation: $$2\pi = 360^\circ \tag 7$$ or $$2\pi = 360\times \left (\frac{1}{360} \right ) \tag 8$$ We can divide the equation with 360 to get how many arcs has $\frac{1}{360}$ of the circle. Thus, we obtain the following: $$\frac{2\pi}{360} = \left (\frac{1}{360} \right ) \tag 9$$ or more readable $$\frac{\pi}{180} \text{ arcs}(\text{rad}) = 1^\circ \tag{10}$$ This means that $\frac{1}{360}$ of the circle has $\frac{\pi}{180}$ arcs. Now we can apply the same technique to get how many parts has one arc. We divide $(8)$ with $2\pi$ and we get: $$\frac{\cancel{2\pi}}{\cancel{2\pi}} = \frac{360}{2\pi}\times \left (\frac{1}{360} \right )$$ or $$1 \text{ arc}(\text{rad}) = \frac{180}{\pi}^\circ$$ Finally for angle $\theta$ measured in radians and angle $\alpha$ measured in degrees we have: $$\theta = \frac{\pi}{180}\alpha$$ and $$\alpha = \frac{180^\circ}{\pi}\theta$$

I would like to know if I am wrong somewhere, because this is only from my point of view. And the last thing, I want to know the meaning of this part of the article on wikipedia about angles: link.

The length of the arc s is then divided by the radius of the arc r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen): $$\theta = k\frac{s}{r}$$

What is the purpose of $k$ here?

• @J.M : Look at my edits and you'll learn how to use \text{} and \tag. ${}\qquad{}$ – Michael Hardy Apr 8 '14 at 1:46
The scaling constant $k$ is equal to $1$ when angles are measured in radians, to $180/\pi$ when in degrees, to $2/\pi$ when in right angles, etc.
• Something about measuring in right angles: Wouldn't k be equal to $\frac{\pi}{2}$? – LearningMath Apr 10 '14 at 16:56