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Radians as ratio of arc length to radius

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality. Created by Sal Khan.

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Video transcript

- [Instructor] What we're going to do in this video is think about a way to measure angles. And there's several ways to do this. You might have seen this leveraging things like degrees in other videos, but now we're going to introduce a new concept. Or maybe you know this concept, another way of looking at this concept. So we have this angle ABC and we want to think about what is a way to figure out a measure of angle ABC. Now one way to think about it would be, well, this angle subtends some arc. In this case it subtends arc AC. And we could see if this angle were smaller, if its measure were smaller, it would subtend a smaller arc. The length of that arc would be smaller. And if the angle were wider or had a larger measure, if it looked something like that, then the arc length would be larger. So should we define the measure of an angle like this as being equal to the length of the arc that it subtends? Is this a good measure? Well some of you might immediately see a problem with that because this length, the length of the arc that is subtended, is not just dependent on the angle, the measure of the angle, it also depends on how big of a circle you're dealing with. If the radius is larger then you're gonna have a larger arc length. For example, let me introduce another circle here. So we have the same angle measure, the central angle right over here, you could say angle ABC is still the same, but now it subtends a different arc in these two different circles. You have this arc right over here. Let's call this arc DE, and the length of arc DE is not equal to the length of AC. And so we can't measure an angle just by the length of the arc that it subtends if that angle is a central angle in a circle. So we can get rid of that equality here. But what could we do? Well you might realize that these two pis that I just created, you could kind of say pi ABC and pi DBE, these are similar pis. Now we're not used to talking in terms of similar pis, but what does it mean to be similar? Well, you have similarity if you can map one thing on to another, one shape on to another, through not just rigid transformations, but also through dilations. And in this situation, if you were to take pi ABC and just dilate it by a scale factor larger than one there's some scale factor where you would dilate it out to pi DBE. And what's interesting about that is if two things are similar that means the ratio between corresponding parts are going to be the same. So for example, the ratio of the length of arc AC to segment, the length of segment BC is going to be equal to the ratio of the length of arc DE to the length of segment BE. So maybe this is a good measure for an angle. And it is indeed a measure that we use in geometry and trigonometry and throughout mathematics and we call it the radian measure of an angle. And it equals the ratio of the arc length subtended by that angle to the radius. We just saw that in both of these situations. So let's see if we can make this a little bit more tangible. Let's say we had a circle here and it has a central point. Let's just call that point, I don't know, F. And then let me create an angle here. And actually I could make a right angle. So let's call that F. And let's see, call this point G, and let's call this H. And let's say that the radius over here is two meters. And now what would be the length of the arc subtended by angle GFH? Well this could even be one 1/4 the circumference of this entire circle, the way that I've drawn it. So the entire circumference, you could write it here, the circumference is going to be equal to 2 pi times the radius, which is going to be two pi times two meters so it's going to be four pi meters. And so if this arc length is 1/4 of that this is going to be pi meters. And so based on this arc length and this radius, what is going to be the measure of angle GFH in radians? Well we could say the measure of angle GFH in radians is gonna be the ratio between the length of the arc subtended and the radius. And so it's going to be pi meters over two meters. The meters, you could view those as canceling out. Which equals pi over two, and pi over two what? Well we would say this is equal to pi over two radians! Now one thing to think about is why do we call it radians. It seems close to the word radius. And one way to think about it is when you divide this length by the length of the radius you're figuring out how many of the radii is equivalent to the arc length in question. So, for example, in this situation one radii would look something like this. If you took the same length and you just went around like this, so you can see it's going to be one point something radii. And that's why you could also say it's one point something radians. If you took pi divided by two you're going to get a little bit over one. You're gonna get one point, I don't know, O, seven, something.