Arc length from subtended angle: radians
About this video
This video finds the length of an arc when the central angle is given in radians. It uses the meaning of a radian: an angle of 0.4 radians intercepts an arc that is 0.4 radii long, so with a radius of 5 units the arc is 0.4 x 5 = 2 units.
You'll be able to
- explain what it means for an angle to measure a certain number of radians.
- find an arc length by multiplying the angle in radians by the radius.
- keep track of units when converting from radii to units of length.
Key words
- radian
- An angle measure where an angle of 1 radian intercepts an arc exactly one radius long.
- arc
- A part of the circumference of a circle.
- radius
- The distance from the center of a circle to any point on the circle; plural radii.
- intercept
- An angle intercepts an arc when the arc lies between the two sides of the angle.
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.
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