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Arc length from subtended angle: radians

Sal finds the length of an arc using the radius and the radian measure of the angle subtended by the arc. Created by Sal Khan.

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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.