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# Arc length as fraction of circumference

Video transcript

- 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.