 Hi, I'm Zor. Welcome to Inuzor Education. I continue explaining what kind of elements and objects we will be dealing with in solid geometry. We have covered a few different objects before, like lines, planes, cylindrical surfaces, prisms. Today's topic is cylinders. So I will just explain what cylinders are, what kind of elements cylinders have, etc. I suggest you to watch this lecture from Unizor.com website because it has side notes, which basically can be used as a textbook. Alright, so this is an introductory to cylinders, just explaining what cylinders are. Basically, no properties, no theorems will be discussed today. Alright, so cylinders. Let's imagine that we have a plane. So this is the horizontal plane, that's how I draw it. And let's consider that on this horizontal plane I have a circle. Well, it looks like oval, but basically it's only because we are looking from a side. Now let's consider we have another plane, which is parallel to this one. So this one will be called alpha, and this one will be called beta. Now we will consider a line D, which is not parallel to these two. Now I'm not really going into the details of what is parallel planes and what is the parallel planes and lines, etc. These are topics which will be discussed in details. Right now I'm just relying on your intuitive understanding that the planes are parallel if they do not have an intersection, as well as the plane and the line and the straight line. Alright, so now I have this straight line and I have this circle, let's call it C. It has a center, oval, and it has a radius, let's say R. Now what I'm going to do right now is I'm going to construct a cylindrical surface. We know what that actually is. I had the whole lecture about what is a cylindrical surface. If you don't remember, go back a couple of lectures where it's explained. So I'm going to construct a cylindrical surface using this circle as a director's and this straight line as a generator's, which basically means I will draw a straight line through each point on the surface on this circle, parallel to D-line. So this line, this line, this line, this line, that's how it will be. Now these lines will intercept the parallel plane B somewhere and this would be the intersection. Now without proof, I will actually prove it in the future, but right now I just wanted to say that without the proof, on that plane parallel to the alpha plane, my cylindrical surface will cut another circle, which will be congruent to this one, so just different center O' but the same radius R. So if these lines are forming a cylindrical surface, then if this base is a circle, this base will also be a circle. So these are two base planes. These are two base circles. Now this is my side surface and basically whatever these surfaces are forming is a cylinder. So a cylinder consists of cylindrical surface in between these two base parallel planes and two base circles. Let's call it bottom and top, something like this. They're base circles, top circle and bottom circle, top base and bottom base. So that's basically it. This is a cylinder. Now to be more precise, this is a circular cylinder because this is a circle. Now in theory, you can obviously consider circular surface which is formed by directories which is not a circle, but let's say an oval ellipse or parabola or anything basically. It will be still some kind of a cylinder in between these two planes. However, most likely we will not consider all these cases. Most likely all our cylindrical experience will be with the cylinders which are circular, which means the top and bottom bases are circles. Also there is another important characteristic. Now this line D, which is generatrix, I told it's not parallel to these two planes. However, one particular case when it's perpendicular is important because if these lines all are perpendicular, then we will call it a right cylinder, well in this case right circular cylinder. And let me tell you that in most of the cases if I'm just saying cylinder, I will mean right which means these are perpendicular to the plane and circular which means these are circles cylinder. So I will just abbreviate the word cylinder I will use in most of the cases, unless it's specifically described. In most of the cases when I'm saying cylinder I will mean a right circular cylinder. Now what else is important? Another terminology thing. This line from O to O' is called an axis. An axis of the cylinder. Well primarily because the cylinder can be considered as a result of rotation of this rectangle. Let's call this point A and this point A' So rectangle O A A' O' So if you will rotate it around O O' it will actually make up a cylinder. In this case obviously right circular cylinder. Well that's basically all I wanted to say. Let me just check. One more terminology. If it's not a right cylinder when it's really tilted it's called oblique. And one of the examples I mean obviously very rough example is the tower of Pisa which is kind of oblique. It's tilted, right? But again I think we will rarely consider oblique cylinders. Most likely it will be right circular cylinders which I will just call cylinders. Well that's actually it. And that's all my short introduction into an object called cylinder which we will consider in many details in the future lectures which will be dedicated specifically to theorems and properties, characteristics of cylinders. So that's it for today. Thank you very much and good luck.