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

About radians:

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: $$\theta = \frac{l}{r} \tag 4$$

About degrees:

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?

Thanks in advance.

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  • $\begingroup$ It is not entirely clear what you are asking.... $\endgroup$ – copper.hat Apr 8 '14 at 1:40
  • $\begingroup$ I am asking for a review of my understanding of angles and for a second opinion :) $\endgroup$ – LearningMath Apr 8 '14 at 1:41
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    $\begingroup$ @J.M : Look at my edits and you'll learn how to use \text{} and \tag. ${}\qquad{}$ $\endgroup$ – Michael Hardy Apr 8 '14 at 1:46
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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.

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  • $\begingroup$ Thanks for your answer. Can you say something about my reasoning above? I don't know how correct it is or can I think of it that way. $\endgroup$ – LearningMath Apr 9 '14 at 22:13
  • $\begingroup$ Something about measuring in right angles: Wouldn't k be equal to $\frac{\pi}{2}$? $\endgroup$ – LearningMath Apr 10 '14 at 16:56
  • $\begingroup$ @J.M : No. Look at the standard definition of radians. $\endgroup$ – Michael Hardy Apr 10 '14 at 18:41

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