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)?
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.