Hi everyone. Welcome to our lecture on the length of a circular arc. What we're going to do in this lecture, give you a nice circle, and whenever I give you a circle where we're going to define the center, we're going to give you a radius that will make up a circle. And in this particular case I'm interested in the length that an object travels when moving around the perimeter of the circle. Sometimes better known as the circumference. So you can imagine some object that starts on the perimeter, call it point A. And it moves along. Maybe it's orbiting something, who knows? But it moves along to point B but in the circular path, okay. That distance is denoted by the letter S often and it's called arc length. It's called arc length. So the question is we want to find the measure of this arc. Sometimes you see it written as the measure, so little M of A to B. So we name the two points where it's going. Let me draw a little arc over A and B To signify that is the measure from A to B. So what is pretty neat about this, and perhaps you can let your imagination run wild as to the physical applications of this, is that, if you draw the two radii, and remember they're the same. So they connect the center of the circle to point A and the center of the circle point B. You get two sort of raise or line segments, that create an angle. And what's really neat about this is that you can use the angle to find the distance traveled. And now you can imagine measuring distances in space and watching how far things are in and all the applications this is. So you can use angles to find arc length. And it's a pretty beautiful sort of experiment to go through and it comes down from the simple observation that I'm going to take a part to the whole of the angle and equate it to the part to the whole ratio of the arc length. So right here I just wrote part over whole equals part of a whole, you may say, well, that's dumb, that's obvious, sort of. The left side I really want to think about it in terms of the circumference. So from the reminder circumference is the perimeter of the circle, and on the right side I really want to think about it in terms of the angle, the angle measure. And this is the key here since we're doing some calculations, I want this to be in radiance, I want this in radiance. I could do it in degrees but were too fancy for that. Okay, so what do I mean by that? So the whole the, part of the circumference, the piece that I'm after is the mystery, right? Find s find the measure of arc AB. So that's my numerator and the entire circumference we know that, the entire circumference the formula for circumference is 2pi r. So that's another formula we've used that a few times now. So just keep that handy in case you forgot. The formula for the circumference is 2pi r, and the radius is given. And this will be equal to the now the part of the angle. So here's going to hold theta, the part that's cut out of the circle versus the whole piece. So what is the entire measure of an angle one lap around? Well in radiance, that's 2pi. And we have this beautiful proportion. This equality of ratios. We like this a lot. We do a little bit of algebra, we move it over. So we move it over to 2pi r to the other side. And this will cancel with theta over 2pi. So the two and the pi they cancel, and you're left with s equals or arc length equals r times theta. It's a really simple, beautiful formula. The only catch, the only catch, the only catch is that thetas and radiance. This comes from the formula that we used. So this version of the formula, you can always convert it to degrees I guess if you want to but usually do things in radiance. So this is our little formula, one of the things that we're going to get to and that's going to be sort of a theme of this class is where these formulas come from, how they are derived. I don't want them to feel like magic. Like okay here it is, memorize it. If you memorize something, you're going to forget. If you understand where it comes from, you start to see patterns of how formulas are created. You can derive them if you need to, the understanding leads to better memorization. So, this is what I want you to see. I want you to see where these formulas come from. So s equals r theta. And again, a little watch out theaters and radiance. Let's just do an example. So New York city and Bogota, Colombia, they are approximately on the same Meridian. And so, let's find the distance between them. So you can look this up, we're going to approximate a little bit here. The latitude of New York city was about 40.5 degrees North of the equator. So if you imagine the equator kind of going right through Equator here, and you take its angle from the center, it's about 40.5 degrees North. And if you look up Bogota latitude then it's about 4.6 degrees north closer to the equator. So it has less of an angle. So that's pretty good. So these are your angles form from the equator. Okay, what is the angle sort of between the cities? Okay, if I get that, then I'll be able to find s which will be my distance between them. Remember we're going to use the formula s equals r theta. So what is r in this equation? Well, r in this equation is the radius of the earth. And lucky for us that is known, radius of the earth is about to get around in here, but there 3900 miles, okay? So I'm going to use the known latitude, I'm going to use the known value phrase here. So I want to find the distance between New York city and Bogota, okay? So let's try to find that angle. I all ready know what r is, the goal is to find the angle between the cities. So what is theta, you gotta remember from the center of the earth whatever that is, I know that the angle from New York is 40 and the angle from Bogota is 44.6. So the thing I want is the difference between them, so 40.5 degrees minus 4.6 degrees. That's an easy subtraction equation. So that's just 35.9 degrees. And now I'm ready to plug everything in. So my distance in this case here, which is actually an arc length which is s is r theta, tried to formula one more time. R is your radius of the earth. So that's 3900 miles, you can look that up, times my angle. All right, so now be careful here, I'm going to put 35.9 degrees but I hope you're yelling at me as you see this. Remember the warning, the angle has to be in radiance, so not in radiance. Watch out for that. If we just multiply this together, we've got the wrong answer. I have to convert this. I can do it all in one shot. So let's do 3900 miles times 39.5. Remember the conversion factor, to convert I want my degrees on the bottom, so I have 180 degrees, so the degrees cancel and I just put pi radiance upstairs. Now I can work this out. And you can check, if you want to just do this separately and not within the equations, you can certainly do so. It's about 0.6265, somewhere around there. The radiance, but if you work this out and again grab a calculator and check me on this. After rounding a little bit, you get 2481 miles, okay. This is a word problem with real units. So make sure you put miles on this thing, don't leave a blank. It's not cats or donuts or cats per donut, whatever that is. It's truly miles coming from the fact that our units for the radius of the earth was given in miles. And you can measure these things using your stick and a sun in the shadow and get these things and figure out this distance. To show you the ultimate source of knowledge, is this actually, is this math actually correct? How do we know this is true? 2481, well I looked it up on Google. Google tells me it's 2.86, so that little rounding that I did there, that is pretty darn close. So we should feel pretty good about that calculation. One other thing we can get to by just sort of studying angles is the area of a sector, a sector given a circle center and two radii. A sector is the area formed between two radii, so it's not quite a triangle but it has a triangle feel to it, it's got two straight sides and then a rounded arc length for its base here. So this is called a sector. If you want the visual of course it's a slice of pizza, but we're going to use the same sort of pattern to find this thing. We're going to study the angle theta that comes from the sector. So we'll label some things here. So r is known, we'll call the angle theta. And we're going to use the same sort of philosophy. I want to look at what is the part to the whole and I want to equate that too part of the whole. So I want equal ratios. On the left side of course though, since I'm trying to find the area, we're going to look at the area of the sector. So we're going to look at areas, and on the right side over here we're going to look at, well, I need to use the angles somehow. So let's keep using the angles here. So these knowledge of angles really has nice applications. Okay, so what I'm after is what is the area of this sector? We'll call it capital A for area and who knows what that is. We'll find it. So the part that I'm after is capital A, the unknown, the whole circle, who remembers the formula for the area of a circle? Do you remember, pi r squared? If you said that, give yourself a little pat on the back. Okay, so there's my part that I want over the entire area. And on the angle side, the angle that this sector cuts out is called theta and then the entire one lap around is equal to 2pi. So have a nice proportion here. Move some things around. So multiply both sides by pi r squared and then ties it by theta over 2pi. We have some cancelation with the pi's, of course not as much as last time. And you normally see this written with the constant out front. So one half and then r squared times data. So let's put all together as our nice new formula. So the area of a sector of a circle with radius R and whose angle is theta central angle sometimes these are called is one half r squared theta. It's a really nice formula to have and the same warning as before since we used radiance here. So put a little asterisk here, theta must be in radiance, must be in radiance. Watch out for that, they give it to you in degrees. Just practice converting would turn it into radiance when you use this formula. So equals one-half r squared theta. Same flavor, same sort of formula recipe of where this came from. All right, so now of course real world math at its finest. Let's find the area of a slice of pizza. We'll just do the top part. Hello beautiful. All right, here we go. So, let's make up some numbers here. Let's say that the angle, so you get out your little commentary. You measure the angle here at the very vertex of the pizza. Let's give this angle 60 degrees. Okay, so I have our angle be 60 degrees. And let's say we measure our pizza from the vertex to the crust and that'll be our radius and so we'll say r equals 10 centimeters. So I want to know what is the area of a sector. Area of our pizza. Can you think of a better example than pizza for a sector? I certainly can't. Okay, here we go. So area is, let's write the formula down. You could practice one half r squared theta. Of course our warning is that theta must be in radiance. Here, I have that data is 60 degrees. No, no, no, no, no, no good. You could work out the formula if you want. Pause the video and try to remember what 60 degrees is in terms of radiance. But of course 60 degrees is, how do I think about 60 degrees, 60 degrees is one-third of 180. 60 degrees 120, 180. So one 180's pi. So I cut it in three-five of a three. It's interesting I kind of go forward to walk backwards. You can certainly use the conversion factor and get it. But in radiance it's pi over 3 radiance. That's the number that we want to use in our formula. So we have one half. We have our radius which was carefully measured to be 10 centimeters, and our angle of 60 degrees. But we're going to use radiance of course. When we work this out, you can grab a calculator if you want, you can give your answer in terms of pie but sometimes it's better they usually ask for a decimal. This turns out to be 5.24 centimeters squared. If you're wondering where the square centimeters come from, the fact that the radius was given in centimeters dictates your units and against this is an area not a measure of the length. If you want the total area you want to square the units, so centimeters squared or square centimeters. Both are perfectly fine. I think the big takeaway here is one. Just make sure that when you do this with the calculator, that your calculator is in radiance. If you're doing working with angles or that you convert things clearly. Watch a rounding on your final decimal and always, always, always include your units. All right, wonderful application of applied math. Great job on this video. We'll see you next time.
