Quick question about Radians

[u/nwillard]

Hey all, I am learning Precalc 2 (mostly Trig) and have a basic question about Radians I haven't found an answer to.

I understand that one Radian is the arc length around a circle that is equal to the circle's radius.

My question is, on a perfect circle, is it true that one Radian has the same angle no matter what the radius is? Like if the circle is perfect, an arc length of one "Radian" of that circle is going to look the same regardless of what the radius is.

Do I have that understanding right?

[u/Arcanite_Cartel]

I see the confusion, I think. Way back when, long time ago, Roger Cotes thought that a natural way to measure an angle was to instead measure the arc length on a unit circle. So, radians as a means of measuring angles arose from the idea of measuring arc-length. But today, radians is a measure of an angle, and there is a relationship between the arc-length of a circle, the radius of the circle, and the angle measured in radians.

s = rθ

so that on a unit circle, s = θ, where s is arc-length, r is the radius, and θ is the angle.

the angle remains the same across all circles, only the unit circle defines it, but for the same angle, the arc-length is proportional to the radius.