# How is an angle's degree determined?

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)?

• See MSE 720924 etc. – Benjamin Dickman Jul 12 '18 at 6:11
• 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 6:15
• 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 6:48
• @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 6:55
• " 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 7:04

## 1 Answer

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.