Hi everyone, welcome to our lecture on the length of a circular arc. What we're going to do in this lecture, is give you a nice circle. Whenever I give you a circle, we're always 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. You can imagine some object that starts on the perimeter, call it point A, then it moves along. Maybe it's orbiting something, who knows, but it moves along to point B but in the circular path. That distance is denoted by the letter S often, and it's called arc length. The question is we want to find the measure of this arc, sometimes you'll see it written as the measure, so little m, of A-B. So we name the two points where it's going and we draw a little arc over A and B to signify that is the measure from A-B. 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 to point B, you get two raise or line segments that create an angle. What's really neat about this is that you can use the angle to find the distance traveled. Now you can imagine measuring distances in space and watching how far things are and all the applications. You can use angles to find arc length and it's a pretty beautiful 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 whole ratio of the arc length. Right here I just wrote part over whole equals part over whole. You may say, well, that's dumb, that's obvious. The left side, I really want to think about it in terms of the circumference. Friendly reminder, circumference is the perimeter of the circle. On the right side, I really want to think about it in terms of the angle measure and this is the key here, since we're doing some calculations, I want this to be in radians. I could do it in degrees, but, well, we're too fancy for that. What do I mean by that? The part of the circumference, the piece that I'm after is the mystery. Like 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 2Pir. 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 2Pir and the radius is given. This will be equal to 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, or what is the entire measure of an angle one lap around? Well, in radians, that's 2Pi. 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. We move over the 2Pir to the other side, and this will cancel with Theta over 2Pi. The two and the Pi may 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 is that Theta is in radians. This comes from the formula that we used. This version of the formula, you can always convert it to degrees, I guess if you want to, but we usually do things in radians.. This is our little formula. One of the things that we're going to get to, and that's going to be 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. Understanding leads to the better memorization. This one, I want you see where these formulas come from. S equals r Theta, and again, the little watch out, Theta is in radians. Let's just do an example. New York City and Bogota, Colombia, they are approximately on the same meridian. Let's find the distance between them. You can look this up. I'm going to approximate a little bit here. The latitude of New York City was about 40.5 degrees north of the equator. If you imagine the equator going right through Ecuador here, and you take its angle from the center, it's about 40.5 degrees north. 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. That's pretty good. These are your angles formed from the equator. Okay. What is the angle between the cities? 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 rTheta. what is r in this equation? Well, r in this equation is the radius of the Earth. Lucky for us that is known, the radius of the Earth is about, to get around in here, but 3,900 miles. Okay? I'm going to use the known latitudes on them, use the known value first here. I want to find the distance between New York City and Bogota. Okay, let's try to find that angle. I already know what r is. The goal is to find the angle between the cities. Like what is Theta?You got to 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 4.6. The Theta I want is the difference between them, so 40.5 degrees minus 4.6 degrees, that's the easy subtraction equation. That's just 35.9 degrees. Now I'm ready to plug everything in. My distance, in this case here, which is actually an arc lane, which is S, is rTheta, through the formula one more time, r is your radius of the Earth. That's 3,900 miles, you can look that up, times my angle. All right, now be careful here. I'm going to put 35.9 degrees, but I hope you are like yelling at me. As you see this, remember the warning, the angle has to be in radians, so not in radians. Watch out for that. If we just multiply this together, we'll get the wrong answer. I have to convert this. I can do it all in one shot, so let's do 3,900 miles times 39.5. Remember the conversion factor. To convert, I want my degrees on the bottom, so I have a 180 degrees so the degrees cancel. I just put Pi radians upstairs. Now I can work this out. 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, radians. But if you work this out and again, grab a calculator and check me on this. After rounding a little bit, you get 2,481 miles.Okay? This is a word problem with real units. Make sure you put miles in this thing. Don't leave them blank. It's not cats or donuts or cats per doughnut, whatever that is. It's truly miles coming from the fact that our units for the radius of the Earth was given in miles. You can measure these things using your stick and the sun on the shadow and get these things and figure out this distance. To show you the ultimate source of knowledge, 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 into radii, a sector is the area formed between two radii. It's not quite a triangle, but it has a triangle field to it. It's got to straight sides and then a rounded arc length for its base here. 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 pattern to find this thing. We're going to study the angle Theta that comes from the sector. We'll label some things here, r is known, we'll call the angle Theta. We're going to use the same philosophy. I want to look at what is the part to the whole. I want to equate that to part to 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. We're looking at areas. On the right side over here, we're going to look at, well, I need to use the angle somehow. So let's keep using the angles here. So this knowledge of angles is 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? If you remember, Pi r squared. If you said that give yourself a little pat on the back. Okay, so there's my part divide by one 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 good old two Pi. So I have a nice proportion here. Move some things around. So multiply both sides by Pi r squared and then ties it by Theta over two Pi. We have some cancellation with the Pi's, of course not as much as last time, and you normally see this written with a constant out front. So one-half and then r squared times Theta. 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 angles. Sometimes these are called is one-half r squared Theta. It's a really nice formula to have and same warning as before, since we used radians here. So put a little asterisk here. Theta must be in radians, must be in radians. Watch out for that they give it to you in degrees, just practice converting would turn it into a radiance when you use this formula. So A equals one-half r squared Theta, same flavor, same sort of formula recipe of where this came from. Alright, so now, of course, real-world math at its finest. Let's find the area of a slice of pizza, we will just do the top part. Hello, beautiful. Alright, here we go. So let's make up some numbers here. Let's say that the angle, so you get out your little compass here, you measure the angle here, at the very vertex of the pizza. Let's give this angle 60 degrees, so I have our angle is 60 degrees and let's say we measure our pizza, the vertex to the crust, and that'll be our radius 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, good practice one-half r squared Theta. Of course our warning is that Theta must be in radians. Here I have that Theta 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 radians. But of course, 60 degrees is, how do I think about 60 degrees? Sixty degrees is one third of 180, 60 degrees 120, 180. So 180 is Pi, so I cut it in three Pi over three. It's interesting I kind of go forward to walk backwards. You can certainly use the conversion factor and get it. But in radians, it's Pi over three radians. 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, what we're going to use, radians of course. When we work this out, you can grab a calculator if you want to. You can give your answer in terms of Pi, 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 again this is an area and not a measure of a length, 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 a calculator, that your calculator is in radians if you're doing work with angles or that you can convert things clearly. Watch your rounding on your final decimal and always, always, always include your units. Alright, wonderful application of applied math. Great job in this video, we'll see you next time.
