 Hi, I'm Zor. Welcome to Unisor Education. I would like to start a new topic. This is cylinders, a continuation of the part of this course which is dedicated to solid geometry. So, we'll talk about the cylinders and primarily about two characteristics of the cylinder, area and the volume. Well, I do suggest you to watch this lecture from the Unisor.com website because it contains not only the reference to the video itself, but also notes which can serve as a textbook. And then there is a whole functionality which is built into this website. It involves enrolling students into particular courses and then exams, etc. So, cylinders. Well, with a little stretch of imagination, you can view this as a cylinder. Sorry about drawings. Let me just remind you a little bit. Cylinder basically is defined as a geometrical object which is bounded by its side surface which is a cylindrical surface. And I'll talk about what cylindrical surface actually is. And two planes, top and bottom. Also, what's important about this cylindrical surface which is a side surface of this object. Now, cylindrical surface is formed by, if you remember, a straight line called generatrix which is moving parallel to itself along some kind of a curve in the direction of the directories in the 3D space. In this case, directories is a circle. A circle on some kind of a plane. So, if the directories is a circle, then generatrix generates this round cylindrical surface. And this generatrix is supposed to be perpendicular to the plane where the directories, the circular base of the cylinder is located. Now, as far as these two planes, top and bottom, they also should be parallel to the plane where the directories is located. Or you can think that the bottom, let's say, base is a directories or the top base, which is a circle, is a directories. So, that's how the whole cylinder is built. It's a cylindrical surface with a straight line as a generatrix with a circular directories. And the generatrix is supposed to be perpendicular to the plane where the directories is located. So, that's how the cylinder is built. Now, what are parameters which characterize this circle? Well, actually there are two parameters. And they completely define the cylinder. These parameters are the radius of a circle and its height or altitude. So, this is a distance between two bases, between the top and the bottom base, distance between two parallel planes, if you wish. And the R is radius of the base. Both of them, obviously, are the same. Well, there are a few statements which I just made which probably require, in the very rigorous sense, some proof, for instance, how can I prove that if I have a cylindrical surface with a circular directories, how can I prove that these two planes will intersect according to the circles which are exactly the same as the directories? These are kind of trivial theorems and I'll probably do it in some other lecture just as an exercise. Or give it an exam, for instance, that would be fine. So, today I would like to talk about the area of the surface and the volume of the cylinder. Based on these two parameters which presumably completely identify the cylinder. Alright, let's talk about the surface first. Now, the surface of this object contains the side surface which is this rounded cylindrical surface and two bases. Well, two bases are simple. There are two circles. So, the area of the base is pi R square. The area of two bases is 2 pi R square. Okay, now how about this cylindrical surface, what's its area? Now, I would like to approach this from intuitive level rather than from the level of rigorous proof. Intuitively, it is obvious that if you cut our cylinder along one of the generators and just open it up, it will be a rectangle. Now, the height of this rectangle will be exactly h, obviously. Now, what would be the length of this horizontal segment? Well, let's just think about it. If you cut this and open it up, obviously the length of this circumference would be exactly this. Right? So, it would be 2 pi R. Now, this is kind of an intuitive understanding of this and I think I would be quite satisfied with this. Obviously, there is some more rigorous proof and I will talk about rigorousness a little bit further when I will talk about the volume. It probably would require some usage of the limit theory. But for now, I think this intuitive understanding of what is the area of the side cylindrical surface, I think it's quite sufficient actually to basically state that intuitively it's obvious that this particular circumference will open up in this segment and the height would be equal to this. And again, since we have said from the very beginning that the generator is perpendicular to the plane where the director is located, right? So, that's why this is rectangle. That's why this angle is right, 90 degrees. Which means that the area of this particular side surface of a cylinder is equal to 2 pi R H, 2 pi R times H. And if you add them together, you have a full surface of a cylinder which you can factor out 2 pi R R plus H, if you wish. I don't want you to remember this formula, what I do want you to remember is that you can actually cut the cylinder and convert it into the side cylindrical surface and convert it into a rectangle. And then add just two areas, top and bottom base and that would be it. So, I don't remember this formula but as you see I can derive it in like a minute. So, that's basically something which is related to the area. Now, let's talk about the volume. Well, volume is slightly more complex thing in this particular case and I don't think I can avoid using the limit theory in this case. And the way how I'm going to do it is the following. Let's inscribe into the circle which is a base a regular polygon with n sides. Something like this. In this case I have inscribed hexagon. And do exactly the same on the top by just doing this, something like this. So, what I have done is this is invisible and this is invisible, something like this. So, what I did was inscribed the regular polygon with n vertices and then just have perpendicular within the cylindrical surface to this base up until it goes to the top base. And obviously on the top base I will have exactly the same similar actually, congruent n-sided polygon. Now, what happens right now with my cylinder? Now, I have a prism inscribed into a cylinder. Right? Now, let's think about it. What is the volume of this prism? Well, the volume of the prism we know what it is. It's the area of the base times height. Okay. Now, let's increase the number of sides of the polygon which I have inscribed. Let's say instead of this, I will have this. Instead of this, I will have this. So, more and more vertices would be part of this. So, it's more tightly inscribed into a circle. And correspondingly, my prism will be tighter and tighter inscribed into a cylinder. And here, again, intuitively obvious that as number of vertices in this regular polygon at the end goes to infinity, my polygon will be closer and closer to the circle. And my prism inscribed into a cylinder would be closer and closer to the cylinder itself. So, the volume of the prism probably will tend to the volume of the cylinder. Now, what does it mean? The area of the polygon would be closer and closer to the area of the circle, which is pi r squared. The height of the prism is always the same. It's the same height as the height of the cylinder. So, the volume of the prism will be closer and closer to this, which I can actually say that this is probably the volume of the cylinder. So, this is the area and this is the volume. Now, I actually referred to your intuition a couple of times here. This is not rigorous as the true mathematician would probably present it. However, number one, I think for educational purposes, this is even better. However, I would like you to still have a feel of how the rigorous approach would probably proceed in this particular case. Well, let's just take, for instance, this particular volume. Now, to be precise, what I think I would like to do is the following. I have this inscribed prism into a cylinder. I can also have circumscribed prism. So, I will circumscribe a polygon around this circle. So, it's bigger than the circle. And build a prism out of this bigger polygon. Now, if the first prism was inside the cylinder, the second prism would be outside of the cylinder. So, one prism would be inscribed, another circumscribed. And then, what I would do, I would take the limit of both my inscribed prism and circumscribed prism as number of sides goes to infinity. And the limit probably would be the same. I mean, I have to prove it, obviously. But it's kind of obvious that the tighter prism you have around the cylinder, it will still probably be, as a limit, it will go to a certain number. And if you have an inscribed prism, it will go probably to also the number. And if these limits are the same, remember there was a theorem. If you have three different variables, and these two are going into the same, this is variables which are indexed by n. And as n is increasing, these two have the same limit. It means this is also the same limit. Now, in our case, this would be the value of the cylinder. This is the value of the inscribed prism, and this would be the volume of circumscribed prism. And if their volumes are going into the same limit, it means that my volume of the cylinder is exactly this limit. So, this is how it can be done with the volume. And by the way, very similarly, we can do the same with the area of the side surface. How? Again, let's inscribe the rectangle, not rectangle, the polygon, some insided polygon inside the base of the cylinder. And do basically this prism construction. Now, the side area of the cylinder would be around this prism. Now, what is the area of the side surface of the prism? Well, that's some of these rectangles, right? Each side of this right prism is a rectangle. So, if I will summarize the areas of these rectangles, it would be what? This side of the polygon times h, another side times h, etc. So, I will have to add all these sides together and multiply it by h to get the area, right? So, let me just write it down. So, if my insided polygon has the length of the side a, then I will have to have n times h, right? a times h would be the area of one particular face of this prism. And n is number of these prisms, which is exactly the same as this. And what is n times a? n times a is a perimeter of this n-sided polygon inscribed into a circle. And as n goes to infinity, obviously this perimeter will go to the circumference of the circle. So, that will go to 2 pi r. Again, it needs to be proven in exactly the same fashion. I can do an inscribed polygon and circumscribed polygon outside of a circle and check their perimeters and make sure that their perimeters, as n goes to infinity, goes to the same limit and that limit will be, therefore, the length of the circle. So, this is just the road how we can proceed with more rigorous proof of whatever I was just saying. But again, I would quite frankly prefer you to go by your intuition. And intuition shows, basically, that the area, the cylinder, it looks like a prism with infinite number of small faces on the side, if you wish. And therefore, every formula which is related to the prism, like, for instance, the volume of the prism is area of the base times height. It is actually applicable to a cylinder as well. So again, view a cylinder as a prism with infinite number of infinite small side faces. That would be probably a proper description of that. So, that's probably it for today. I would like you to read the notes for this lecture. They are presented at Unizor.com. And try to, just in your head, try to really go through all these logical assumptions with limits, etc., which will allow you to build your intuition in this particular case. So, your intuition should really lead you to a conclusion that there is no much difference between the prism and the cylinder. And whatever the properties of the prism are, more or less, general for all the prisms, those are probably true for cylinders as well. Well, that's it for today. Thank you very much and good luck.