Radians as ratio of arc length to radius | Circles | High school geometry | Khan Academy

What we're going to do in this video is think about a way to measure angles. 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. 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 going to 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. 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 pies that I just created, you could kind of say pi ABC and pi DBE. These are similar pies. Now, we're not used to talking in terms of similar pies, but what does it mean to be similar? Well, you have similarity if you can map one thing onto another, one shape onto another, through not just rigid transformations, but also through dilations.

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. 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 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; maybe this is a good measure for an angle. 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, you know, F, and then let me create an angle here; and actually I can make a right angle, so let's call it F. And let's call this point G, and let's call this H.

Let's say that the radius over here is 2 meters. Now, what would be the length of the arc subtended by angle GFH? Well, it's going to be one-fourth the circumference of this entire circle, the way that I've drawn it. So, the entire circumference, I could write here, the circumference is going to be equal to 2π times the radius, which is going to be 2π times 2 meters, which is going to be 4π meters.

So, if this arc length is one-fourth of that, this is going to be π 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 going to be the ratio between the length of the arc subtended and the radius. And so, it's going to be π meters over 2 meters; the meters, you could view those as cancelling out, which equals π/2.

And π/2 what? Well, we would say this is equal to π/2 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 figure 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 π divided by 2, you're going to get a little bit over one; you're going to get one point 07 something.