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Subtended angle from arc length

CCSS Math: HSG.C.B.5

Video transcript

A circle has a circumference of 20 pi. It has an arc length of 221/18 pi. What is the central angle of the arc in degrees? So they're asking for this one. So this is the arc that they're talking about, that's 221/18 pi long. And they want to know this angle that it subtends, this central angle right over here. So we just have to remind ourselves that the ratio of this arc length to the entire circumference-- let me write that down-- the ratio of this arc length, which is 221/18 pi, to the entire circumference, which is 20 pi, is going to be equal to the ratio of this central angle, which we can call theta, the ratio of theta to 360 degrees if we were to go all the way around the circle. This will give us our theta in degrees. If we wanted it in radians, we would think of it in terms of 2 pi radians around the circle, but it was 360 degrees since we're in degrees. Now we just have to simplify. Now the easiest thing is just to multiply both sides times 360 degrees. So let's do that. So if we multiply the left-hand side by 360 degrees, we get 360 times 221 times pi over-- let's see, we have 18 times 20 times pi. And on the right-hand side, if we multiply it by 360, we are just left with theta. So we really just have to simplify this now. Pi divided by pi is going to be 1. 360 divided by 20, well, it's going to be the same thing as 36/2, which is the same thing as 18. And 18 divided by 18 is 1. So this all simplified to 221 degrees. Theta is 221 degrees.