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 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 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 sometimes you take each of the endpoints of the arc go to the center of the circle go 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 2 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 all right so let's 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 I I how do i how do I think through this and we just have to remind ourselves what radians mean what radians mean if an arc subtends an angle of 2 radians that means that the arc itself is 2 radius this is is long so this right over here so let me let me let me make this a little clearer so if this if the radius is R if this radius do this I already use that color if this rain I have trouble switching colors alright if this radius is length R then the length if this if this angle is 2 radians then the arc that subtends it is going to be 2 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 2 pi times the radius or you could say it's 2 pi radii 2 pi radiuses 2 pi radio is the correct way to say it so what fraction is it it's 2 radii it's 2 radii over 2 pi radii over 2 pi radii twos cancel out hours cancel out and so it is 1 pi I guess you could say it is 1 over pi of the total circumference