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# Arc length from subtended angle: radians

Video transcript

P is the center
of the green arc. So this is p right over here. The measure of angle
P is 0.4 radians, and the length of the
radius is 5 units. That's this length
right over here. Find the length
of the green arc. So just to kind of
conceptualize this a little bit, P, you can imagine, is the
center of this larger circle. And this angle
right over here that has a measure of 0.4 radians, it
intercepts this green arc right over here. In order to figure this
out-- and actually, I encourage you to pause
this video now and try to think about this
question on your own. How long is this arc,
given the information that they've given us? Well, all we have to
do is remind ourselves what a radian is. One way to think about a radian
is if you look at the arc that the angle intercepts, which
is this green arc, if you think about its length, the
length of this green arc is going to be 0.4 radii. One way to think
about radians is if this angle is
0.4 radians, that means that the arc
that it intercepts is going to be 0.4 radii long. So this length we could
write as 0.4 radii. Or "radiuses," but "radii"
is the proper term. Now, we don't want our
length in terms of radii. We want our length in terms
of whatever units the radius is, these kind of 5 units. Well, we know that each
radius has a length of 5, that our radius of the
circle has a length of 5. So this is going to
be 0.4 radii times 5-- and you know they just call
it units right over here. I'll put it in quotes because
it's kind of a generic term-- 5 units per radii. So the radii cancel out. We're left with just the
units, which is what we want. So 0.4 times 5 is 2. So this is going
to be equal to 2. So just as a
refresher again, when the angle measure
in radians, one way to think about it is the
arc that it intercepts, that's going to be
this many radii long. Well, if each radius
is 5 units, it's going to be 0.4 times
5 units long, or 2.