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# Subtended angle from arcĀ length

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.