Arc length as fraction of circumference

- Let's say that I have a circle. My best attempt to draw a reasonably perfect circle. So, there you go, not too bad, it's a little bit of a hairy circle but you get the idea. So, this is a circle, this is the center of the circle, and let's say that I have an arc along this circle. So, I'll do the arc in green. So, I have an arc that is part of the circle, and it subtends an angle, so that's my arc. Right over there, and it subtends an angle, and the angle that it subtends, so what I mean subtends, you take each of the endpoints of the arc, go to the center of the circle, go to the center of the circle just like this, and so it subtends angle theta, right over here, so it subtends angle theta, and let's say that we know that angle theta is equal to two radians. So my question to you is what fraction of the entire circumference is this green arc? What fraction of the entire circumference is this green arc? And like always, pause the video, and give it a go. (laughs) All right, so let's think through it a little bit. So, you might say well how do I know that, I don't know what the radius of this thing is, I don't, how do I think through this? And we just have to remind ourselves what radians mean, what radians mean. If an arc subtends the angle of two radians, that means that the arc itself is two "radiuseseses" long. (laughs) So, this right over here, let me make this a little clearer, so this, if the radius is r, if this radius is, I already used that color, if this radius... I have trouble switching colors (laughs) all right. If this radius is length r, then the length, if this angle is two radians, then the arc that subtends it is going to be two radiuses long, so this length right over here, is two radiuses. Now, what fraction of the entire circumference is that? Well, the entire circumference, we know, we know this from basic geometry, the entire circumference is two pi times the radius, or you can say it's two pi radii, two pi "radiuseses", (laughs) two pi radii is the correct way to say it. So, what fraction is it? It's two radii, it's two radii, over two pi radii, over two pi radii, twos cancel out, rs cancel out, and so it is one "pith", (laughs) I guess you could say, it is one over pi of the total circumference.