Radians conversion

I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance

I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance

I'm having a really hard time understanding why you multiply by pi/180 to get degrees. I completely don't understand why this is used, and how you arrive at this. I know that pi radians = 180 degrees but I don't understand how this is used to convert into radians/degrees. I originally thought this was used by rearranging but even then it doesn't make sense to me. This is really stressing me out as its a really fundamental point within this topic and I don't understand it! All help is really appreciated so thank you in asvance

With radians, we have decided to define the circle as $2 \pi$ units rather than 360 and so a particular angle in radians is made into a fraction of a circle by dividing by $2 \pi$.

I think that this is a rather back-to-front way to express it. For a circle of radius $r$, a radian is usually defined as the amount of angle subtended by an arc of length $r$ at the centre.

So imagine measuring a piece of string of length $r$ and wrapping it around the circumference of the circle, then draw radius lines from the centre to the ends of the arc - the amount of angle between those two radii is 1 radian. Then, we can say that since the circumference is $2\pi r$, then we can fit exactly $2\pi$ arcs of that size around the circumference, and hence since each of those arcs subtends 1 radian, we get $2\pi$ radians of angle around a circle.

You are of course quite correct. I am sure that the OP will find your post illuminating My primary objective was to give them some tools to cope with the difficulties they were describing. What you say is accurate and reveals a great deal about the origin of radians rather than addressing the student's problem.

The 180 only arises because of the cancellation of the 2 in $2 \pi$ with 360.

Which is why tau is much better.

Help me plz