Why do we find the arc length when given an angle in degrees the way we do?

Question

I have a very stupid question. In order to find the arc length when given an angle in degrees, my textbook said we must use this formula: $\frac{angle (degrees)}{360o}$ * 2$\pi$r = Arc length

It didn’t explain why this formula works or where they got it from, however. Something is bothering me about it: it makes too much sense.

I get why the formula you use to find an angle measure in radians works.

$\theta$ = $\frac{s}{r}$

It just comes right out of the definition of a radian.

In order to get the first formula I mentioned, I know you can just convert to degrees by multiplying both sides by $\frac{360o}{2\pi r}$. You would do this because radians are defined in terms of arc length, so you can find arc length using their defintion. Degrees are not, so you must convert in order to find arc length.

However, the first formula makes intuitive sense; the fraction of the angle times the circumference should yield the corresponding arc measure. If we get it by converting units, it seems like it shouldn’t make sense. So my question is this: are there some postulates or definitions that make the first formula make sense? Is there another way we could get that formula without just using the one for an angle in radians and converting that utilizes fundamental definitions?

Thank you in advance!

• Dan: It's simple algebra 2ドル\pi \text{ rad} = 360°$ means that $\frac{\pi \text{ rad}}{180°} = 1,ドル giving a conversion factor between the two units of angle measure. How exactly is this confusing for you?
• Moon: @Dan I understand that part. It’s just that if the formula for finding the arc length given an angle in degrees is just the formula for finding the arc length given an angle in radians converted using a conversion factor, why does it still make intuitive sense? I can’t find any definitions or postulates backing up the intuitive reasoning behind the first formula. Am I overthinking it too much?
• fleablood: How can it not?! An arc length is a proportion a circle's circumference and an angle is a proportion of a circle's rotation. If you cover a certain proportion of a one full revolution the arc length is going to be exactly that proportion of the circumference of a circle. How on earth can that not be intuitive and obvious? Why would you expect that to not be true?
• fleablood: Okay... A circle is perfect an uniform. If a circle has a total circumference of $C$ and you go $k$% of if, then the arc length will be $k$% of $C$ and as 360ドル^\circ$ is the angle for a full circle, going $k$% of $C$ you will have gone an angle of $k$% of 360ドル^\circ$. I can't see why anyone would be upset that that makes sense. Of sense.
• fleablood: " If we get it by converting units, it seems like it shouldn’t make sense." See. this sentence makes absolutely no sense to me. If you have two thirds of a gross and you convert a gross to 12 dozen you should have 2/3 of 12 (eight) dozen. And if a dozen is 12ドル$ units you have eight times 12 units. Why would you expect that to NOT make sense. Your arc length is two thirds of the circumference of a circle that is measured as 2pi radians, that should be two thirds of the angle measured in degrees. You should be HAPPY that it makes sense. I can comprehend thinking that is SHOULDN"T

Answer 1

No beed to mention radians. All you need to know is:

1. For a given radius, the arc length is directly proportional to the angle.
2. A full circle is 360ドル$ degrees (this is essentially the definition of "degree").
3. The arc length of a circle of radius $r$ is 2ドル\pi r$ (this is essentially the definition of $\pi$).

So if you have an arc with an angle of $d$ degrees and radius $r$, that is $d/360$ of a circle of radius $r$, and the arc length is that same fraction of the arc length of the circle, namely $$\frac{d}{360} \times 2 \pi r = \frac{d \pi r}{180}$$