 The arc length of a circle can be computed by the formula s equals r times theta, where s is the measure of the arc in question, the length of the arc. r is the length of the radius of the circle in play here. And then theta is the measure of the angle that gives us the associated arc here. But that angle measurement must be in radians. It's essential that the angle be measured in radians. This comes from the fact that the length of the arc is directly proportional to the angle in play. And that's where we derived radian measure in the first place. And as such, when issues like, well, the measurement of the angle needs to be in degrees or some other type of angle measurement system, we have to remember we have to convert from radians to degrees or degrees to radians, depending on which direction we're going. So suppose that the arc of a circle has a length of approximately 31 feet. And so we're gonna say specifically that the, just to make the arithmetic a little bit easier, let's say that the arc is 10 times pi, which is approximately 31 feet. If the circle has a radius of six feet, then what is the measurement of the angle? And we're gonna do that in degrees. Well, we're gonna use our formula right here where we have that s equals 10 pi. We have that r is equal to six. So again, these are all measured in feet. And so theta is the one thing we don't know. And so taking our arc length formula right here, we can divide by r and we get that theta is gonna equal s over r. That is we get 10 pi over six, which as a fraction six and 10, both have a common factor of two. And so canceling out that two, we get five pi over three. And this would be the measurement of the angle in play right here. But again, this is gonna be a measurement in radians. If we want the measurement to be in degrees, because that's a very common angle measurement system there. To do this in degrees, we have to convert from radians into degrees. And basically when you have a radian measured, you don't want the pi there. So kind of cancel out the pi. To get into degrees here, you're gonna take the five pi over three radians. And you need a time set by 180 degrees over pi, like so. And simplifying this thing, of course the pi's cancel out. That was the convenience of using 10 pi as opposed to 31. But we also need to simplify the rest of the fraction here. Three does go into 180, and that gives us 60 degrees right there. 60 degrees times that by five, you're gonna end up with 300 degrees as the measurement of the angle right there. And so if your angle measure is not in radians, you do have to convert it over into whatever angle measurement you're using. Most likely if it's not in radians, it's gonna be in degrees. I say most likely though, because there are perhaps other less common ways of measuring angle measures. And so in a trigonometry course, you often see a question similar to the following. The minute hand of a clock is 1.2 centimeters long. And it's not very big. When you say clock makes me think, is it actually a watch? But we'll call it a clock just for the sake of it, right? How far does the tip of the minute hand move in 20 minutes here? So thinking about our analog clock, we have our minute hand right here. And so we're saying, okay, this thing is 1.2 centimeters long. But then, so that gives us a measurement of radius, right? We think of the circle, the minute hand, that would be our radius there. The radius is equal to 1.2 centimeters, okay. So what is it that we're actually being asked to find? How far does the tip of the minute hand move in 20 minutes? So if we think of this minute hand as spinning, right? It's gonna spin around the clock clockwise, of course. That's why we call it clockwise. And so as it spins some distance, this is how far the minute hand would move. So there's like this initial minute hand and this terminal minute hand. Okay, we're trying to measure this distance right here. This is measuring an arc length S. But to find the arc length S equals our theta, we need to know the radius, which we have, but we also need to know the angle measure. And theta here, what's our clue about the angle measure here? The clue here is time. So on a analog watch or clock, 20 minutes elaps. And that's our measurement of angle. That's sort of an unorthodox way of measuring an angle, but it makes sense. I mean, do fighter pilots not say things like, oh, you got a bogey at six o'clock, right? That's referencing a clock, which is a different way of referencing a circle. So we can give positions of a circle based upon measurements of time. And so that's what's happening right here. So the thing to remember is if you're gonna measure an angle using time here, how much is in one complete rotation, right? So when it comes to one revolution, one rotation around the clock, in terms of radians, we know this is two pi radians. In terms of degrees, we know this is 360 degrees. This is actually how we found conversion between radians and degrees. Well, if we're measuring the circle using minutes, there's gonna be 60 minutes in an hour, which one hour would be one rotation of the minute hand on your clock. And so this is how we're gonna relate things together here. We get that two pi radians is equal to 60 minutes, all right? And so if we wanna convert between minutes and radians, we could do exactly that. Now, we don't have to do the entire 60 minutes. Of course, if we're doing 20 minutes here, that's actually one third of a revolution, one third of a revolution of the clock. And so we need to take one third of two pi here. So that tells us that 20 minutes is actually equal to two pi thirds in terms of angle measure right there. So just recognizing that 20 minutes is a third there. Of course, we could also continue on with what we have right here. If you want to convert between radians and minutes or something, you could divide both sides by 60 and you get that two pi over 60 radians is equal to one minute. That fraction does simplify. You can then times both sides by 20, all right? Upon doing that, you'll get 20 minutes is then equal to, well, 20 goes into 63 times and you end up with this two pi thirds in radians again. So we've converted from minutes to radians. And then once we've done that, we then can compute the arc length there. You're gonna take S to be 1.2 centimeters times the radian measure, which is two pi thirds. Like so, given that the radius is already measured in a decimal there, 1.2, we're just gonna approximate this and round it to the nearest 10th place. And so taking 1.2 times two pi over three, put that in your calculator, you would end up with, if you want the precise answer, it's actually gonna be four pi over five. But like I said, a decimal approximation is appropriate here. You're gonna get 2.5 centimeters. So the important thing when you work with arc length formula here is that whenever, whenever you have an angle measure other than radians, you have to convert it.