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.

00:00:15In our previous class we'd talked about circumference and circles.
00:00:21And we were solving the relationship between the plane of a line and circumference of a circle.
00:00:29That was mostly a geometrical construction and today we'll take a look at calculating numerical tasks.
00:00:35But the first main obstacle will be to figure out the reason and purpose of the circumference.
00:00:44So as a headline, write... Length of the circumference.
00:01:03It does not matter, it doesn't matter today. Length of the circumference and very often we use another term, perimeter of a circle.
00:01:23Length of the circumference, perimeter of a circle. So when you see these two terminologies, you'll know it's the same.
00:01:34I 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.
00:01:44Why am I changing the name as such? Because if you recall how to determine, for example... what-
00:02:09Perimeter of what.
00:02:11Perimeter of a square, you'll be able to remember that, yes?
00:02:13Four times A.
00:02:14Yes, so in this case we're talking about the perimeter. Formula. Paul?
00:02:26(inaudible) Four times A.
00:02:29Four times A. Make sure no one makes a mistake. If you have this type of figure...
00:02:46Two times A. Symbol S.
00:02:50That's debatable.
00:02:52The area.
00:02:53What are we trying to figure out.
00:02:54The perimeter.
00:02:55If we're looking for the perimeter then the symbol will be?
00:02:58Correct. Perimeter, which we mark as O. And you know?
00:03:04O equals two times A plus two times B.
00:03:15We can continue with this. Nicolie?
00:03:24A to the second power plus B to the second power.
00:03:26I think you overdid it with that one, wouldn't you say?
00:03:29A to the second power times B.
00:03:31What 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.
00:03:38And this side? You've mentioned that you take both sides twice and add it up. I can do that as well. Excuse me?
00:03:53Two times A plus B.
00:03:58I can add it up, because there are two pairs. We add A plus B and multiply... yes, so...
00:04:04By two.
00:04:05By two correct. So, two times A plus B. You may use this formula.
00:04:10When you start studying Algebra, you'll need to know this formula.
00:04:15You'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.
00:04:23It will become handy. So, you guys know how to define it when it comes to these forms.
00:04:28You also know how to figure out a triangle, yes?
00:04:30O equals A plus B plus C.
00:04:33Yes, when you have three sides, you just add up the three sides of a triangle.
00:04:40And now what about the circumference. Perimeter of a circle or the distance of a circumference, that's a difficult question.
00:05:07How to go about solving it. People have been solving these problems as early as 2,000 years ago.
00:05:30Mr. Archimedes figured out the perimeter of a circle formula. Do you have an idea? John?
00:05:43You somehow draw a square.
00:05:47May I come up to the board?
00:05:48You may.
00:06:06Good job. We know the perimeter of a square. We can figure that out. I have to assume the circumference, that this drawing which-
00:06:19This drawing which John outlined corresponds to what...
00:06:28What does this correspond to?
00:06:31No, no. To what?
00:06:32Secant of circumference.
00:06:34That is correct as well. What is this...
00:06:40Excuse me?
00:06:41Length of the square.
00:06:42Yes, so this will be the length of the square. And what does this blue segment in the circumference mean?
00:06:52Linear equation.
00:06:53It's not a linear equation, it's a segment.
00:06:56Diameter. Correct, diameter. So, John is suggesting a diameter of the circumference, which- or to compare it with the side of a square.
00:07:11He's said that D equals to A and from that he can figure out the perimeter of a circle.
00:07:27What's being left out when he substituted the length of the circumference with the perimeter of a circle. Is that correct?
00:07:33It's not accurate.
00:07:34It's not accurate. Simply, it's not accurate. The shape is different. I just can't replace it as we did in this example.
00:07:45So, I'll do it more accurately. Which more accurate method should I use?
00:07:50I would try to divide it into four segments.
00:07:57With what?
00:07:58Through center S.
00:07:59Through center S and then what, once you divide it up.
00:08:01Well in order to find out the radius... to know the radius of the circumference, I multiply the radius by four.
00:08:09When I will have three centimeters times four.
00:08:14Well, why four times for instance.
00:08:15Because, I'll divide it into four pieces.
00:08:18Yes, but...
00:08:20That's the same thing as if I divided it in eight pieces and that would be the same.
00:08:24Well, you're making a mistake because that would still be this, yes.
00:08:26You 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.
00:08:37Could I draw the square in the inside?
00:08:39Yes, that would work. That's a good idea. Draw it inside. Inscribe it.
00:08:44The difficulty is that when I inscribe the square, compared to how simple it is here, one side (inaudible) equals the diameter.
00:08:53So, 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.
00:09:07Then the diameter is by what? The diameter in the square is by what? The diameter in the square is?
00:09:30Diagonal. The diameter in the square is diagonal. We would have a great difficulty determining the area of a square.
00:09:44Not that it wouldn't work. We would figure it out. After all... well...
00:10:01I think I complicated it for you. Would you be able to do something else?
00:10:08The perimeter in this case... the perimeter of a square would be a problem in this case why because you don't know...
00:10:13The lengths.
00:10:15Sides. Now you could calculate it, since you know the root, you ought to be able to calculate the area from the square.
00:10:22And 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.
00:10:30So, I would for example use an octagon.
00:10:34Octagon. You will not be able to determine the sides. Where can you easily determine the sides.
00:10:40No. Not the triangle. Stanley?
00:10:46Hexagon. 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.
00:11:00The radius. And what ever comes out of that... is what?
00:11:17Hexagon. What is the length of one side.
00:11:21Radius of the circumference.
00:11:24Well, I used the radius. I measured it with the radius. The length of one side of the hexagon is the length of the radius.
00:11:35It still isn't accurate.
00:11:37It still is not accurate. So how would you define it more precisely?
00:11:42Wait a minute. Hexahedron.
00:11:43Dodecagon, octagon...
00:11:46Which one?
00:11:48Dodecagon, 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.
00:12:04Then you can do a icositetragon...
00:12:11Would that work?
00:12:12Up to enneacontakaihexagon. You would not be able to figure it out not even the twelfth angle.
00:12:21But more of the angles...
00:12:22But the more angles we have, the more likely it will start to look like a circumference.
00:12:26And then the perimeter can be applied to the length of the circumference. That's the formula Mr. Archimedes created.
00:12:38Okay? We can write a note; Archimedes used for his calculations the ninety-sixth angle. The ninety-sixth angle.
00:13:14You can read it in your books that this was in the year 2300 B.C. Three hundred years before Christ.
00:13:24He 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,
00:13:35is arriving to one thing, that the perimeter of a square has four diameters, the perimeter of a hexagon, Philip?
00:14:04Six of what.
00:14:05Diameters. Radius.
00:14:07Radius. Six radius. And if you recall, you can see it from this picture, diameter and radius are in what condition? Diameter and radius?
00:14:29The radius is half of the diameter.
00:14:31The radius is half of the diameter, so here I can use the diameter, once again I'll write, three times two times R.
00:14:44And the two times R is...
00:14:50Diameter, correct. So, it came out to be three times D. From this calculation you can see one thing, that the perimeter of the circle,
00:15:01the length of the circumference is somewhere between the fourth diameter and the third diameter. It is not very accurate, that's correct.
00:15:12Mr. Archimedes has tried as I've mentioned, he used the polygon, and he came out with a more accurate result.
00:15:20This ought to be enough for our lecture. We'll write up a conclusion to this matter.
00:15:38Note, the length of the circumference depends on... what do you think it depends on?
00:16:01Or radius, yes. It depends on the diameter. You know from your seventh grade the proportionality. I could write that it's simply proportional.
00:16:32Diameter. The length of the circumference is a proportional diameter. Do you also remember the equation of that?
00:16:43K equals three.
00:16:48That part is correct. What about the actual formula.
00:16:54Y equals K over X.
00:17:03K times X. If I say proportional, then we recalled the formula of K times X. Take a look at our discovery here.
00:17:14Once again we have diameter here and here... and in front of that is a number or a constant.
00:17:18So, I can consider, that the length of the circumference will be proportional as to K times... what did we say, diameter.
00:17:30The difficulty is that we still don't know the letter K. That we don't know the constant.
00:17:41But 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.
00:17:55Is it going to be four?
00:17:56No, No.
00:17:57That 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?
00:18:05Between the two.
00:18:06Between the two. So, the letter A will be between the two. This is not an accurate number.
00:18:19As you've mentioned earlier, if we were to use other polygons, then we would be getting a more accurate number.
00:18:26And since we're not able to figure out the accurate calculations, we must consider a formula, which will be enough for this sample.
00:18:50The number of three point fourteen or 22 divided by seven.
00:19:11These numbers are rounded-off. If you take a look at the graph tables in your book, you'll see it on page 54.
00:19:35All 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.
00:19:52And 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.
00:20:03You may try it on your calculator but you may not have the option on this particular calculator.
00:20:07I'll give you calculators later on, which does not have the option either.
00:20:10You'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.
00:21:03The Pi is named after Ludolf. Why is it named after Ludolf? Does anyone know? Anyone?
00:21:33Ludolf Van Ceulen, he was from Holland and he calculated the same thing as Mr. Archimedes but around the year 1600.
00:21:54This information is only for the curious, around the year 1600 he calculated it to several numbers nearest the tenth decimal.
00:22:00Approximately 30 numbers nearest to the tenth decimal. Which is not very practical but he found out one thing, that the number,
00:22:13this number looks like it could be replaced with a fraction, but you would find out that 22 divided by seven,
00:22:19and 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.
00:22:31If you were to punch in the number you would come out with several different... Michael, that no same number is repeated.
00:22:41It doesn't have a periodic cycle nor is it divisible, not even with a remainder of zero, so no same remainder is repeated.
00:22:47So, it's a number which cannot be formulated with a fraction. Note, Pi cannot be formulated with a fraction. Okay?
00:23:21That 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...
00:23:32Either the 22 divided by seven or Pi as three point fourteen to the nearest tenth.
00:23:54Can't be formulated with a fraction. What does that mean? If you were to take a ruler and measure a radius or diameter.
00:24:09You will not... in this system, you will not be able to exactly determine the length of the circumference. To measure it.
00:24:18Not that you're not capable of calculating it. You take a circumference mark a point on it and you let it roll, yes?
00:24:33When this point reaches the equation then here you have the measurement of the circumference.
00:24:40But if you measured the radius correctly, I'm sure you will not be able to measure with the same gauge.
00:24:47The radius and the measurement of the circumference is in this relation.
00:24:51Okay, in order to wrap this up, did we discover a direct proportion? Let's write that instead of K, Paul please complete it.
00:25:19Pi, pi, pi.
00:25:20And 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.
00:25:46How should I write it up?
00:25:47Two times R.
00:25:48Two times R. So, I'll write O equals Pi times two times R. It's common to write the numbers before the letters.
00:25:59So, I'll write O equals two times Pi times R. Both of the formulas are used.
00:26:09Those of you that can't remember it, can look it up in table graphs. You'll use a certain formula depending on the situation.
00:26:47Calculate the length of the circumference, given... Can you Teresa and Petra distribute the calculators please?
00:27:02Calculate the length of the circumference. Letter A, D equals two point four decimeters. Letter B, D equals one point forty-five centimeters.
00:27:49The assignment requires only multiplying and you have only decimal numbers in order not to spend too much time on calculation.
00:28:04That's the most important principal of this assignment. You must calculate the length of the circumference if you know the radius.
00:28:13The mathematical steps are similar like in Physics. Make a note Stanley.
00:28:22Perimeter equals.
00:28:23What's already written there. Write it into the formula and calculate it. Come up here Martin.
00:28:42You'll pick a formula that is suited for this problem. Why are you changing it?
00:29:12Diameter and then I can change it to...
00:29:15You may, but why would you do that since you have the formula? Yes.
00:29:34We will convert to decimeters.
00:29:37Come on Martin. That may be possible but... I still don't understand the logic of why you switched the numbers.
00:29:52I know it like this. (inaudible)
00:29:59One more time?
00:30:01I believe I have to know the diameter.
00:30:04I would write it in the same spot where the diameter is. You're correct, it's not significant, but... that has no significance either.
00:30:13You don't know the number of Pi?
00:30:15I know.
00:30:18Why don't you choose the numbers.
00:30:23Three point fourteen.
00:30:24Three 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...
00:31:01Exactly. Do you all understand? The first thing Martin could have or should have done is, when it's a substitute for a rounded number,
00:31:16then he needs to write the word approximate above the equation. He needs to write approximately here as well.
00:31:22And the next thing is that this result, should be written in the nearest decimal unit.
00:31:29And the answer ought to be in real numbers. Those are just details you should consider. Eric.
00:31:52You should be able to remember the formula. If you're having troubles remembering it, you may look it up on page 44.
00:32:12Do you understand? In time, you'll realize that these exact answers will not be useful.
00:32:21Because as you continue to use these formulas, you'll be rounding it off to the nearest decimal point.
00:32:24If I have a dimension which is to the nearest tenth and we rounded off at the hundredth point. Is it clear? Is it hard?
00:32:47Centimeters. Good, next person, Nik. Just erase the information which we don't need anymore so we can see the problems clearly.
00:33:04You may leave the formula there just erase that over there, yes.
00:33:24He may change the radius to the diameter, he may use it but he can also use the other formula.
00:34:30Six 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.
00:34:48Try to calculate the perimeter. Mike, you're not thinking.
00:35:30I just made a note that if you can cleverly calculate it then instead of using three point fourteen you may use a fraction.
00:35:41But 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.
00:35:50Rounded off, here you must multiply with an approximate number. Approximately seven point forty-five centimeters. Good.
00:36:13That was problem D. And how about problem E?
00:36:23You have (inaudible), that should have been nine point one hundred and six thousandths.
00:36:33Why don't you fix it. You punched in incorrect numbers... let me see. Go ahead calculate it. Six point twenty-eight times... nine point...
00:36:58She made a mistake. She punched in an incorrect number. It was overlooked. Go up Jana and correct it.
00:37:29Good. And once again Simona, since this over here is an approximate number then here you must always write the word approximately.
00:37:36Even though on the first look it looks like a complete number. But the value is approximate. Pi is an approximate value.
00:37:50Do you all understand it?
00:37:53There is nothing to it. These problems will be a bit more difficult. I'm sure you will not have too much trouble with it.
00:38:04Find or calculate the diameter... the diameter of the circumference. The perimeter.
00:38:49Calculate the diameter and in this similar problem, calculate... the radius...
00:39:27You? Go up there and try to calculate it. Calculate the diameter and radius.
00:39:54That's not bad, but this is something we... we haven't mentioned a formula for this.
00:40:04So, you're going to have to in some way figure it out, or you must use the formula which we've discovered
00:40:12And then form it with some substitutions and adjustments, you can come up with an answer.
00:40:17It's not bad but there are three variations from the formula, which we need to list.
00:40:24I can outline it in your grade books, but you will not remember it. We'll use the simplest kind.
00:40:29So, always write that the perimeter is Pi times D, yes. Substitute, as we did in Physics, four point two.
00:40:44Three point fourteen times D, good. Approximate number, yes. And D is approximate, you have to be careful here.
00:40:56The formula. Generally, you had it right. Yes, calculate it and you're done. Do you all understand?
00:41:50This number doesn't work well for you.
00:41:53You calculated it to the nearest tenth point and the result comes out to 10 numbers so you can round off to approximately-
00:42:05Thirty. To the nearest hundredth, yes? One point thirty. Nikolai...
00:42:22Yes, underlined twice. Now Teresa can go up and try it.
00:43:09Good, yes. And the next line, whatever Nikolai wrote over there. What's important is the letter D now. Yes, yes.
00:43:35Calculate it. You will learn these formulas... you'll learn how to work with it and later on we'll study more technical problems.
00:43:51You'll get familiar with it and then we'll cover area. You'll have a complete understanding of the circumference.
00:44:10Teresa, always write approximate number. Good. We'll continue on Monday.
00:44:20Take a rest over the weekend, you don't have any assignments for the weekend. On Monday we'll continue with area.