Trigonometry: Using the Arc Length Formula

good and straight forward

02/01/2011

~ Michael131

This lesson is quick but very good.

• View all learner reviews »

Using the Arc Length Formula
The real power of radian measure actually can be seen right out of the get-go by taking a look at what's referred to as arc length. Now, what is an arc length? Well, it's just a literally a length of an arc, an arc of a circle. So, let's suppose I have a circle and let's suppose it has radius . And suppose I take a look at a little wedge, so I make an angle, and I'm going to call that angle "theta." By the way, a lot of times we use the Greek letter "th" to represent the measure of an angle. So, that's just the Greek letter, th, lower case. Now, you'll notice that if this thing has radius and I sweep out th as the angle, there's an arc here and sometimes it's actually useful to know how long that arc is, not the circumference of the entire thing, but just the length of that little piece right there. So what is the arc length of this arc?
Well, it turns out since we're using radians as the measure of angle, and we know that sort of once around is a complete cycle, so that would be 2, you can actually then find the arc length of any arc for any circle by the following formula. So, the arc length, which is sometimes referred to as "s" for arc length - don't ask me - is actually equal to the radius, measured in whatever, multiplied by th, the angle measured in radians. This will not work is you're using degrees. So, it's a very simple formula.
For example, let's just check this and see if makes sense. What if I go make one complete cycle around? Then the arc I would be spanning would be the entire arc of the circle. So what's the arc length of that? Well, this formula says it's going to be times that angle in radians. Well, once around is 2, so that would be x 2, or 2. But wait a minute. Once around is just the circumference. And that's 2. So, you can see this checks with what we know, that once around that circumference is 2. And that's what this formula says, times 2. And, so therefore, halfway around would be what? It would just be , because this angle would be and I multiply it by . So, in fact, when you measure things in radians, you can immediately start to get lengths of arcs and so forth. As a little teeny application, let's just take a look at the following. Suppose that we have a surveyor. A surveyor is someone that sort of, you know, looks into one of these things and looks out, spies into it and then sort of wants to measure how far away things are. And let's suppose that we have a surveyor and an assistant. Here's the assistant, looks like maybe a sumo wrestler, but really it's the surveyor assistant way over here. And the surveyor measures - by the way, suppose that we know that the assistant is six feet, one inch. So, this surveyor is six feet, one inch. And the surveyor has that thing that sort of measures angles, and this angle right here, it's very, very tiny, it's only a half of a degree. And the question is, how far away is the surveyor from the assistant? So what is this distance right here? Well, how could we answer this question?
Well, let's see, what we could do is say, well, this is such a tiny degree. I mean, half a degree is almost nothing. You know, it's like this little thing here. It's so tiny, that in fact, this distance here isn't going to differ too much by the circle that would sort of go through that point, that arc of the circle. So let's find the length of the arc of the circle and let's assume, in fact, let's not find it, let's assume that it's roughly six feet, one inch, the same as the guy, right? Because if it's so tiny, there's very little difference between that little arc of the circle length and the actual straight person length. So, if we do that, we can go into this formula and we know the angle, we know this arc length - six feet, one inch - and we can then solve for the radius. That would tell us how far these two people are.
Now, what do I do with the one-half degrees? Do I put that in here? Absolutely not. I have to convert that first to radians. So how do I convert that to radians? Well, here's what I always do. I just start from ground zero and say, okay, well, 360 = 2. So, that means that 1° = radians. But I want a half a degree, so I multiply everything through by half and I see that a half a degree would equal radians. Aha! The angle is really in radians. So, we put in here and what do we put in here? Here we put in six feet, one inch, which let's just convert it into inches. That would be 73 inches. So, I see 73 inches equals the radius, which is not known to us at the moment, and then the angle is . So, that means that the radius would equal 360 x 73, all over . And what does that work out to be? Well, if you work that out on a calculator, you see that equals, let's see, 8,365.18 inches. Okay, that's inches, but how many feet is that? Well, if you just divide that by 12, you'd see it equals 697.09 feet. So, just knowing the angle and how tall our sumo wrestler assistant was, we could actually figure out how far they are apart. Now, of course, I'm using the fact here that this angle is very, very tiny. So, there's very little difference between the assistant's height and this arc length. But, suppose that, in fact, the angle was bigger so it was very dramatic. Well, then, in fact, the straight line distance would be a lot different than the arc distance. The arc distance would probably be greater. So, it raises the question, how could I find sort of information like this if I just know straight line distance and I don't want to estimate it, using this arc thing, but I want to somehow do it some other way. Well, it turns out that's where the trigonometric functions and the power of trig will come in, because that's how we're going to finally figure out how to do that kind of thing exactly.
But for now, I hope that this arc length formula gives you a sense of, first of all, how to find the length of an arc if the angle is measured in radians, and also the power of radian measure, the measure of choice by mathematicians and by us together.