# CZ2 PERIMETER OF A CIRCLE

This eighth grade mathematics lesson focuses on the derivation and application of the formula for the perimeter of a circle. It is the second lesson in a unit of work focused on the circumference of a circle. The lesson is 45 minutes in duration. There are 15 students in the class.

Time | Caption |
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00:00:15 | In our previous class we'd talked about circumference and circles. |

00:00:21 | And we were solving the relationship between the plane of a line and circumference of a circle. |

00:00:29 | That was mostly a geometrical construction and today we'll take a look at calculating numerical tasks. |

00:00:35 | But the first main obstacle will be to figure out the reason and purpose of the circumference. |

00:00:44 | So as a headline, write... Length of the circumference. |

00:01:00 | (inaudible) |

00:01:03 | It does not matter, it doesn't matter today. Length of the circumference and very often we use another term, perimeter of a circle. |

00:01:23 | Length of the circumference, perimeter of a circle. So when you see these two terminologies, you'll know it's the same. |

00:01:34 | I may be talking about the length of the circumference and sometimes I'll call it perimeter of a circle and you'll know it's the same thing. |

00:01:44 | Why am I changing the name as such? Because if you recall how to determine, for example... what- |

00:02:08 | Perimeter. |

00:02:09 | Perimeter of what. |

00:02:10 | Square |

00:02:11 | Perimeter of a square, you'll be able to remember that, yes? |

00:02:13 | Four times A. |

00:02:14 | Yes, so in this case we're talking about the perimeter. Formula. Paul? |

00:02:26 | (inaudible) Four times A. |

00:02:29 | Four times A. Make sure no one makes a mistake. If you have this type of figure... |

00:02:46 | Two times A. Symbol S. |

00:02:50 | That's debatable. |

00:02:52 | The area. |

00:02:53 | What are we trying to figure out. |

00:02:54 | The perimeter. |

00:02:55 | If we're looking for the perimeter then the symbol will be? |

00:02:57 | O. |

00:02:58 | Correct. Perimeter, which we mark as O. And you know? |

00:03:04 | O equals two times A plus two times B. |

00:03:15 | We can continue with this. Nicolie? |

00:03:24 | A to the second power plus B to the second power. |

00:03:26 | I think you overdid it with that one, wouldn't you say? |

00:03:29 | A to the second power times B. |

00:03:31 | What do you have to see here? You can see this side, correct? Here you have another one. Yes, same length and parallel to each other. |

00:03:38 | And this side? You've mentioned that you take both sides twice and add it up. I can do that as well. Excuse me? |

00:03:53 | Two times A plus B. |

00:03:58 | I can add it up, because there are two pairs. We add A plus B and multiply... yes, so... |

00:04:04 | By two. |

00:04:05 | By two correct. So, two times A plus B. You may use this formula. |

00:04:10 | When you start studying Algebra, you'll need to know this formula. |

00:04:15 | You'll have a pair of parentheses and be able to calculate it. Or you'll have a formula and will have to figure it out. |

00:04:23 | It will become handy. So, you guys know how to define it when it comes to these forms. |

00:04:28 | You also know how to figure out a triangle, yes? |

00:04:30 | O equals A plus B plus C. |

00:04:33 | Yes, when you have three sides, you just add up the three sides of a triangle. |

00:04:40 | And now what about the circumference. Perimeter of a circle or the distance of a circumference, that's a difficult question. |

00:05:07 | How to go about solving it. People have been solving these problems as early as 2,000 years ago. |

00:05:30 | Mr. Archimedes figured out the perimeter of a circle formula. Do you have an idea? John? |

00:05:43 | You somehow draw a square. |

00:05:45 | Where. |

00:05:47 | May I come up to the board? |

00:05:48 | You may. |

00:05:56 | B:00] |

00:06:06 | Good job. We know the perimeter of a square. We can figure that out. I have to assume the circumference, that this drawing which- |

00:06:19 | This drawing which John outlined corresponds to what... |

00:06:28 | What does this correspond to? |

00:06:29 | Radius. |

00:06:31 | No, no. To what? |

00:06:32 | Secant of circumference. |

00:06:34 | That is correct as well. What is this... |

00:06:38 | Radius. |

00:06:40 | Excuse me? |

00:06:41 | Length of the square. |

00:06:42 | Yes, so this will be the length of the square. And what does this blue segment in the circumference mean? |

00:06:52 | Linear equation. |

00:06:53 | It's not a linear equation, it's a segment. |

00:06:55 | (inaudible) |

00:06:56 | Diameter. Correct, diameter. So, John is suggesting a diameter of the circumference, which- or to compare it with the side of a square. |

00:07:11 | He's said that D equals to A and from that he can figure out the perimeter of a circle. |

00:07:27 | What's being left out when he substituted the length of the circumference with the perimeter of a circle. Is that correct? |

00:07:32 | Michael? |

00:07:33 | It's not accurate. |

00:07:34 | It's not accurate. Simply, it's not accurate. The shape is different. I just can't replace it as we did in this example. |

00:07:45 | So, I'll do it more accurately. Which more accurate method should I use? |

00:07:50 | I would try to divide it into four segments. |

00:07:57 | With what? |

00:07:58 | Through center S. |

00:07:59 | Through center S and then what, once you divide it up. |

00:08:01 | Well in order to find out the radius... to know the radius of the circumference, I multiply the radius by four. |

00:08:09 | When I will have three centimeters times four. |

00:08:14 | Well, why four times for instance. |

00:08:15 | Because, I'll divide it into four pieces. |

00:08:18 | Yes, but... |

00:08:19 | Yes. |

00:08:20 | That's the same thing as if I divided it in eight pieces and that would be the same. |

00:08:24 | Well, you're making a mistake because that would still be this, yes. |

00:08:26 | You for instance talked about a radius. But that's the same concept as if we would discuss diameter. And there is a lot more of them. |

00:08:37 | Could I draw the square in the inside? |

00:08:39 | Yes, that would work. That's a good idea. Draw it inside. Inscribe it. |

00:08:44 | The difficulty is that when I inscribe the square, compared to how simple it is here, one side (inaudible) equals the diameter. |

00:08:53 | So, when I inscribe it, when... should I do it? I don't want to. Do you know why? Because the picture... (inaudible). You're correct. |

00:09:07 | Then the diameter is by what? The diameter in the square is by what? The diameter in the square is? |

00:09:29 | Diagonal. |

00:09:30 | Diagonal. The diameter in the square is diagonal. We would have a great difficulty determining the area of a square. |

00:09:44 | Not that it wouldn't work. We would figure it out. After all... well... |

00:10:01 | I think I complicated it for you. Would you be able to do something else? |

00:10:08 | The perimeter in this case... the perimeter of a square would be a problem in this case why because you don't know... |

00:10:13 | The lengths. |

00:10:15 | Sides. Now you could calculate it, since you know the root, you ought to be able to calculate the area from the square. |

00:10:22 | And from that you could calculate the sides. Yes. But you simply would not be able to figure it out. So you cannot determine the perimeter of the square. |

00:10:30 | So, I would for example use an octagon. |

00:10:34 | Octagon. You will not be able to determine the sides. Where can you easily determine the sides. |

00:10:39 | Triangle? |

00:10:40 | No. Not the triangle. Stanley? |

00:10:45 | Hexagon. |

00:10:46 | Hexagon. Do you remember? You take a radius, mark it here and here. You place the radius here as well and mark it here and here. |

00:11:00 | The radius. And what ever comes out of that... is what? |

00:11:16 | Hexagon. |

00:11:17 | Hexagon. What is the length of one side. |

00:11:21 | Radius of the circumference. |

00:11:24 | Well, I used the radius. I measured it with the radius. The length of one side of the hexagon is the length of the radius. |

00:11:35 | It still isn't accurate. |

00:11:37 | It still is not accurate. So how would you define it more precisely? |

00:11:40 | Hexahedron. |

00:11:42 | Wait a minute. Hexahedron. |

00:11:43 | Dodecagon, octagon... |

00:11:46 | Which one? |

00:11:47 | Dodecagon. |

00:11:48 | Dodecagon, correct. That means I would mark points here. What else? I'm sure you wouldn't be happy with this... this will be a dodecagon. |

00:12:04 | Then you can do a icositetragon... |

00:12:08 | Forty... |

00:12:09 | Tetraconiakaioctagon. |

00:12:11 | Would that work? |

00:12:12 | Up to enneacontakaihexagon. You would not be able to figure it out not even the twelfth angle. |

00:12:21 | But more of the angles... |

00:12:22 | But the more angles we have, the more likely it will start to look like a circumference. |

00:12:26 | And then the perimeter can be applied to the length of the circumference. That's the formula Mr. Archimedes created. |

00:12:38 | Okay? We can write a note; Archimedes used for his calculations the ninety-sixth angle. The ninety-sixth angle. |

00:13:14 | You can read it in your books that this was in the year 2300 B.C. Three hundred years before Christ. |

00:13:24 | He came up with this formula. I will get back to that later. But from this formula which we used with the help of this square and hexagon, |

00:13:35 | is arriving to one thing, that the perimeter of a square has four diameters, the perimeter of a hexagon, Philip? |

00:14:03 | Six. |

00:14:04 | Six of what. |

00:14:05 | Diameters. Radius. |

00:14:07 | Radius. Six radius. And if you recall, you can see it from this picture, diameter and radius are in what condition? Diameter and radius? |

00:14:29 | The radius is half of the diameter. |

00:14:31 | The radius is half of the diameter, so here I can use the diameter, once again I'll write, three times two times R. |

00:14:44 | And the two times R is... |

00:14:48 | Diameter. |

00:14:50 | Diameter, correct. So, it came out to be three times D. From this calculation you can see one thing, that the perimeter of the circle, |

00:15:01 | the length of the circumference is somewhere between the fourth diameter and the third diameter. It is not very accurate, that's correct. |

00:15:12 | Mr. Archimedes has tried as I've mentioned, he used the polygon, and he came out with a more accurate result. |

00:15:20 | This ought to be enough for our lecture. We'll write up a conclusion to this matter. |

00:15:38 | Note, the length of the circumference depends on... what do you think it depends on? |

00:15:57 | Diameter... |

00:16:01 | Or radius, yes. It depends on the diameter. You know from your seventh grade the proportionality. I could write that it's simply proportional. |

00:16:32 | Diameter. The length of the circumference is a proportional diameter. Do you also remember the equation of that? |

00:16:43 | K equals three. |

00:16:48 | That part is correct. What about the actual formula. |

00:16:54 | Y equals K over X. |

00:17:03 | K times X. If I say proportional, then we recalled the formula of K times X. Take a look at our discovery here. |

00:17:14 | Once again we have diameter here and here... and in front of that is a number or a constant. |

00:17:18 | So, I can consider, that the length of the circumference will be proportional as to K times... what did we say, diameter. |

00:17:30 | The difficulty is that we still don't know the letter K. That we don't know the constant. |

00:17:41 | But from this reasoning, we've found out that the K will be located where? Is it going to be three? No, that would be a hexagon. |

00:17:55 | Is it going to be four? |

00:17:56 | No, No. |

00:17:57 | That would be a square. The square of a larger perimeter and hexagon of a smaller perimeter. So, where is the letter A going to be? |

00:18:05 | Between the two. |

00:18:06 | Between the two. So, the letter A will be between the two. This is not an accurate number. |

00:18:19 | As you've mentioned earlier, if we were to use other polygons, then we would be getting a more accurate number. |

00:18:26 | And since we're not able to figure out the accurate calculations, we must consider a formula, which will be enough for this sample. |

00:18:50 | The number of three point fourteen or 22 divided by seven. |

00:19:11 | These numbers are rounded-off. If you take a look at the graph tables in your book, you'll see it on page 54. |

00:19:35 | All the way on top, the terms with the letter Pi. The term Pi. And the constant which we found as letter K in the circumference is marked as letter Pi. |

00:19:52 | And the number in here is marked at the nearest thousand. If you use a good calculator you'll be able to figure out to the nearest million. |

00:20:03 | You may try it on your calculator but you may not have the option on this particular calculator. |

00:20:07 | I'll give you calculators later on, which does not have the option either. |

00:20:10 | You'll be using only the numbers and formulas which we've used in class. We'll mark it with letter Pi, it's marked as Pi and it's called Ludolf formula. |

00:21:03 | The Pi is named after Ludolf. Why is it named after Ludolf? Does anyone know? Anyone? |

00:21:33 | Ludolf Van Ceulen, he was from Holland and he calculated the same thing as Mr. Archimedes but around the year 1600. |

00:21:54 | This information is only for the curious, around the year 1600 he calculated it to several numbers nearest the tenth decimal. |

00:22:00 | Approximately 30 numbers nearest to the tenth decimal. Which is not very practical but he found out one thing, that the number, |

00:22:13 | this number looks like it could be replaced with a fraction, but you would find out that 22 divided by seven, |

00:22:19 | and three point fourteen are similar, but you cannot place an equal sign between them. In the mathematical tables, this number Pi has other decimal places. |

00:22:31 | If you were to punch in the number you would come out with several different... Michael, that no same number is repeated. |

00:22:41 | It doesn't have a periodic cycle nor is it divisible, not even with a remainder of zero, so no same remainder is repeated. |

00:22:47 | So, it's a number which cannot be formulated with a fraction. Note, Pi cannot be formulated with a fraction. Okay? |

00:23:21 | That is why, as I've shown you over there that this value will be used. I can note that Pi is approximately... you'll always write the dot there... |

00:23:32 | Either the 22 divided by seven or Pi as three point fourteen to the nearest tenth. |

00:23:54 | Can't be formulated with a fraction. What does that mean? If you were to take a ruler and measure a radius or diameter. |

00:24:09 | You will not... in this system, you will not be able to exactly determine the length of the circumference. To measure it. |

00:24:18 | Not that you're not capable of calculating it. You take a circumference mark a point on it and you let it roll, yes? |

00:24:33 | When this point reaches the equation then here you have the measurement of the circumference. |

00:24:40 | But if you measured the radius correctly, I'm sure you will not be able to measure with the same gauge. |

00:24:47 | The radius and the measurement of the circumference is in this relation. |

00:24:51 | Okay, in order to wrap this up, did we discover a direct proportion? Let's write that instead of K, Paul please complete it. |

00:25:19 | Pi, pi, pi. |

00:25:20 | And O is pi times D. The only relation we can use. I wrote over there that the diameter has two radius, then instead of D, I'll write, John. |

00:25:43 | (inaudible) |

00:25:44 | Martin. |

00:25:45 | Half. |

00:25:46 | How should I write it up? |

00:25:47 | Two times R. |

00:25:48 | Two times R. So, I'll write O equals Pi times two times R. It's common to write the numbers before the letters. |

00:25:59 | So, I'll write O equals two times Pi times R. Both of the formulas are used. |

00:26:09 | Those of you that can't remember it, can look it up in table graphs. You'll use a certain formula depending on the situation. |

00:26:47 | Calculate the length of the circumference, given... Can you Teresa and Petra distribute the calculators please? |

00:27:02 | Calculate the length of the circumference. Letter A, D equals two point four decimeters. Letter B, D equals one point forty-five centimeters. |

00:27:49 | The assignment requires only multiplying and you have only decimal numbers in order not to spend too much time on calculation. |

00:28:04 | That's the most important principal of this assignment. You must calculate the length of the circumference if you know the radius. |

00:28:13 | The mathematical steps are similar like in Physics. Make a note Stanley. |

00:28:22 | Perimeter equals. |

00:28:23 | What's already written there. Write it into the formula and calculate it. Come up here Martin. |

00:28:42 | You'll pick a formula that is suited for this problem. Why are you changing it? |

00:29:12 | Diameter and then I can change it to... |

00:29:15 | You may, but why would you do that since you have the formula? Yes. |

00:29:34 | We will convert to decimeters. |

00:29:37 | Come on Martin. That may be possible but... I still don't understand the logic of why you switched the numbers. |

00:29:52 | I know it like this. (inaudible) |

00:29:59 | One more time? |

00:30:01 | I believe I have to know the diameter. |

00:30:04 | I would write it in the same spot where the diameter is. You're correct, it's not significant, but... that has no significance either. |

00:30:13 | You don't know the number of Pi? |

00:30:15 | I know. |

00:30:18 | Why don't you choose the numbers. |

00:30:23 | Three point fourteen. |

00:30:24 | Three point fourteen is used as Pi. The fraction is used only when you can reduce it by a fraction. I'll show you that later. Multiply it and... |

00:31:01 | Exactly. Do you all understand? The first thing Martin could have or should have done is, when it's a substitute for a rounded number, |

00:31:16 | then he needs to write the word approximate above the equation. He needs to write approximately here as well. |

00:31:22 | And the next thing is that this result, should be written in the nearest decimal unit. |

00:31:29 | And the answer ought to be in real numbers. Those are just details you should consider. Eric. |

00:31:52 | You should be able to remember the formula. If you're having troubles remembering it, you may look it up on page 44. |

00:32:12 | Do you understand? In time, you'll realize that these exact answers will not be useful. |

00:32:21 | Because as you continue to use these formulas, you'll be rounding it off to the nearest decimal point. |

00:32:24 | If I have a dimension which is to the nearest tenth and we rounded off at the hundredth point. Is it clear? Is it hard? |

00:32:47 | Centimeters. Good, next person, Nik. Just erase the information which we don't need anymore so we can see the problems clearly. |

00:33:04 | You may leave the formula there just erase that over there, yes. |

00:33:24 | He may change the radius to the diameter, he may use it but he can also use the other formula. |

00:34:30 | Six times (inaudible) zero seventy-two, very good. Jana. I also added E and the value of a diameter as a fraction, which is not common. |

00:34:48 | Try to calculate the perimeter. Mike, you're not thinking. |

00:35:30 | I just made a note that if you can cleverly calculate it then instead of using three point fourteen you may use a fraction. |

00:35:41 | But only if it makes sense. Not that this wouldn't work, it will work. But here you must have it exact, here it has to be approximate. |

00:35:50 | Rounded off, here you must multiply with an approximate number. Approximately seven point forty-five centimeters. Good. |

00:36:13 | That was problem D. And how about problem E? |

00:36:23 | You have (inaudible), that should have been nine point one hundred and six thousandths. |

00:36:33 | Why don't you fix it. You punched in incorrect numbers... let me see. Go ahead calculate it. Six point twenty-eight times... nine point... |

00:36:58 | She made a mistake. She punched in an incorrect number. It was overlooked. Go up Jana and correct it. |

00:37:29 | Good. And once again Simona, since this over here is an approximate number then here you must always write the word approximately. |

00:37:36 | Even though on the first look it looks like a complete number. But the value is approximate. Pi is an approximate value. |

00:37:50 | Do you all understand it? |

00:37:52 | Yes. |

00:37:53 | There is nothing to it. These problems will be a bit more difficult. I'm sure you will not have too much trouble with it. |

00:38:04 | Find or calculate the diameter... the diameter of the circumference. The perimeter. |

00:38:49 | Calculate the diameter and in this similar problem, calculate... the radius... |

00:39:27 | You? Go up there and try to calculate it. Calculate the diameter and radius. |

00:39:54 | That's not bad, but this is something we... we haven't mentioned a formula for this. |

00:40:04 | So, you're going to have to in some way figure it out, or you must use the formula which we've discovered |

00:40:12 | And then form it with some substitutions and adjustments, you can come up with an answer. |

00:40:17 | It's not bad but there are three variations from the formula, which we need to list. |

00:40:24 | I can outline it in your grade books, but you will not remember it. We'll use the simplest kind. |

00:40:29 | So, always write that the perimeter is Pi times D, yes. Substitute, as we did in Physics, four point two. |

00:40:44 | Three point fourteen times D, good. Approximate number, yes. And D is approximate, you have to be careful here. |

00:40:56 | The formula. Generally, you had it right. Yes, calculate it and you're done. Do you all understand? |

00:41:50 | This number doesn't work well for you. |

00:41:53 | You calculated it to the nearest tenth point and the result comes out to 10 numbers so you can round off to approximately- |

00:42:01 | Approximately... |

00:42:02 | Thirty. |

00:42:04 | Hundredths. |

00:42:05 | Thirty. To the nearest hundredth, yes? One point thirty. Nikolai... |

00:42:15 | Yes. |

00:42:22 | Yes, underlined twice. Now Teresa can go up and try it. |

00:43:09 | Good, yes. And the next line, whatever Nikolai wrote over there. What's important is the letter D now. Yes, yes. |

00:43:35 | Calculate it. You will learn these formulas... you'll learn how to work with it and later on we'll study more technical problems. |

00:43:51 | You'll get familiar with it and then we'll cover area. You'll have a complete understanding of the circumference. |

00:44:10 | Teresa, always write approximate number. Good. We'll continue on Monday. |

00:44:20 | Take a rest over the weekend, you don't have any assignments for the weekend. On Monday we'll continue with area. |