0

To be as specific as possible I am not asking the following:

  • What is a degree? (Measurement of rotation between two intersecting rays/lines)
  • How much is a degree? ($\frac{1}{360}$th rotation of a circle)
  • How do you measure an angle in degrees with a protractor?

I've Googled and watched several Youtube videos regarding this question and they all say something along the lines of the items I listed.

Similar to how a Radian is found with radius and arc length, how is a degree found with just the information gathered from an angle (i.e the ray length and the arc length)?

David Garcia
  • 5
  • See MSE 720924 etc. – Benjamin Dickman Jul 12 '18 at 06:11
  • So in a real life scenario I just measure the ray and arc and find out if the angle is a Radian or not That means you know how to measure arcs on the unit circle. I can't imagine how In a real life scenario I would determine the measure of an angle in degrees Measure the arc on the unit circle then multiply by $180 / \pi,$. – dxiv Jul 12 '18 at 06:15
  • @dxiv Im unclear of your solution for finding the degree of an angle by multiplying the arc length by 180/pi. I tried it a circle with a circumference of 10, that has an angle that is 180 degrees. The arc length of the angle is 5 if I multiply that by the ratio you stated I get 286.47. – David Garcia Jul 12 '18 at 06:24
  • @BenjaminDickman I appreciate the link. From what I understood it is restating what I said about degrees being arbitrary and they have not direct relationship to a circle or its components unlike Radian measure. Still I don't see what information is needed to measure degrees, for instance, a Radian measure requires finding the radius and arc length. – David Garcia Jul 12 '18 at 06:31
  • @dxiv yes I know that a semicircle's perimeter is composed of pi radii, I was referring to my example of a semicircle with a perimeter of 5 and how to find the degree measure of the semicircle. To my understanding degrees are not defined using pi. – David Garcia Jul 12 '18 at 06:34
  • @DavidGarcia Leave the radius out, since you are only confusing yourself unnecessarily. Consider the unit circle where the radius is $1$ by definition. Then the arcs you claimed you know how to measure all satisfy $\pi ,\text{radians} = 180 ,\text{degrees},$, $2 \pi ,\text{radians} = 360 ,\text{degrees}$ and so on. I don't understand where your difficulty is with seeing that $1 ,\text{degree} = 180 / \pi ,\text{radians}$. It's like saying that you know how to measure straight lengths in inches, but not in centimeters. – dxiv Jul 12 '18 at 06:40
  • 2
    You seem to think arcs can be measured. And you have no problem with measuring when an arc equals a radius. Well measure the length of a full circle. Divide by $360$. Compare your arc to that. If your arc is $\frac x{360}$ of what the full circle would be it is $x$ degrees. In you example the circle has circumference of $10$ and your arc is $5$. So your arc is $\frac 5{10}=\frac {180}{360}$ of the circle. That's it. – fleablood Jul 12 '18 at 06:48
  • @DavidGarcia Degrees and radians are determined by a ratio. That's why they have no dimension. – Oliver Jones Jul 12 '18 at 06:54
  • 1
    @fleablood I may be incorrect but I assumed like a circumference can be "unrolled" into a straight line and measured an arc can too. Is it not possible to measure arcs? Furthermore, your example seems to have cleared up my confusion. So a degree of my circle of size 10 can be found by dividing by 360, so 1 degree of a circle of size 10 has an arc length of 0.027. – David Garcia Jul 12 '18 at 06:55
  • So let's take a circle with circumference $256.82$ miles and your arc length $97.34$ miles. Then a single degree will be $\frac {256.82}{360} = 0.71339$ miles. Your arc is $\frac {97.34}{0.71339} = 136.447$ of these measurements so you angle is $136.447$ degrees. Or your are is $\frac{97.34}{256.82} = \frac {97.34\times 360\frac 1{256.82}}{256.82\times 360\frac 1{256.82}}=\frac {136.447}{360}$ of your circle. So it is $136.447$ degrees. – fleablood Jul 12 '18 at 06:58
  • @fleablood Awesome that explanation really helped. In that scenario you know the circle circumference, lets say you only know the arc length 97.34 and the length between the vertex of the angle and the point the angle intersects the arc. Is it still somehow possible to find the degree measure? – David Garcia Jul 12 '18 at 06:58
  • @DavidGarcia Degrees and radians are completely independent. You can calculate one without knowing anything about the other. – Oliver Jones Jul 12 '18 at 07:02
  • 1
    " Is it not possible to measure arcs?" It is if we say it is. But in real life we have to have some form of flexible measuring tape. Which is no more acceptable to assume than having a protractor. So measuring when an arc equals a radius is no more "natural" then measuring will an arc is $\frac 1{360}$ of the whole circle. – fleablood Jul 12 '18 at 07:04
  • @OliverJones thanks that is what I figured I want to know how to measure an angle in degrees with just the information about that angle (length between vertex and arc, arc length) is it possible? – David Garcia Jul 12 '18 at 07:05
  • @DavidGarcia You don't need a radius or arc length to calculate a degree. 360 degrees is associated with one complete turn (the number 360 is attributed to the Babylonians). 1 degree means you've made 1/360 of a turn.90 degrees means that you've made a quarter turn. Etc. – Oliver Jones Jul 12 '18 at 07:08
  • Of course! If you know what $\pi$ is. Circumference = $2\pi r$ and angle degrees = $\frac{\text{arclength}}{\text{circumference}}\times 360$. – fleablood Jul 12 '18 at 07:09
  • Radians and degrees both measure the same thing. A full rotation. A degree is $\frac 1{360}$ of a rotation. And it is the same for every circle. A radian is $\frac 1{2\pi}$ of a rotation. Why that irrational fraction? Because a radian is how much of a turn is required to have the arclength equal the radius. That's practical. What is the practical equivalent of a degree. A degree is how much of a turn is required to have the arclength equal $\frac 1{360}$ of the circumference. – fleablood Jul 12 '18 at 07:14
  • thanks a lot guys, really. You probably guessed I'm a newbie at this subject from my possibly "obvious" questions. I appreciate the detailed responses. I feel better now that I understand the fundamentals of each measure. I'll compile the comment responses into an appropriate answer. Thanks again! – David Garcia Jul 12 '18 at 07:20

1 Answers1

0

If you have convinced yourself that a Radian is how much of a turn is required so that the resulting arc length is equal to the radius of the resulting circle...

then a Degree is how much of a turn is required so that the resulting arc length is equal to $\frac 1{360}$ of the circumference of resulting circle.

The identities:

$C = 2\pi r$.

$arc = r\times rad$.

$arc = \frac {degrees}{360}\times C$

$rad = \frac {arc}{r}$.

$degree = \frac {arc}{C}\times 360$.

$degree = \frac {360}{2\pi} rad = \frac {180}{\pi} rad$.

$rad = \frac {\pi}{180} degree$.

Of course, you are assuming arc length, radius and circumference is known. Often the arc length is not known but the proportion of a rotation (whether you consider a rotation to be $2\pi$ radians or $360^\circ$) is known.

fleablood
  • 124,253