 Hi, I'm Zor, welcome to a new Zor education. This lecture is also an introductory lecture to one particular aspect of solid geometry. I will just introduce you to pyramids, basically definition and elements of the pyramids. This is part of a few lectures where I'm just introducing the concepts of solid geometry. I'm not talking about properties, formulas, etc., just explaining what pyramid actually is all about. Now this lecture is part of the Advanced Maths for Teenagers course, which is presented on Unizor.com and that's where I suggest you to watch this lecture from because the site contains notes for the lecture which basically explains exactly the same material and you can always refer to the notes to find out what was in the lecture. Okay, so pyramids. Now I have already explained before about conical surfaces. These are surfaces which are formed by a certain number of straight lines which connect each point of some curve called directories to a fixed point which is called an apex. Now what is the pyramid? It's a special type of conical surface. So to introduce it I just have to explain what kind of a curve lies in the base and that basically defines the pyramid. So I assume that our directories, the curve which is in the base of the pyramid, is some kind of a polygon on a plane. So there is a plane and there is a polygon on this plane. Well, in this case I was trying to put some kind of hexagonal, doesn't really matter what kind of a polygon. So let's imagine that we have a polygon on some plane alpha and this polygon will be used as a directories of the conical surface. Now we choose some apex outside of this plane A and connect each point with an apex. So we will have some kind of a conical surface which contains like two parts before the apex and after the apex. Okay, so this is the beginning of my constructing the pyramid. What we will take from this conical surface is only the part between the apex and the plane where the base is lying. So these are not part of the conical surface which I am considering and these are not. So the conical surface is cut from the top by the apex and from the bottom by my plane alpha. So this part of the surface actually forms the sides of the pyramid and the polygon itself which is in the plane alpha, A, B, C, D, F in this particular case, constitutes another part of this pyramid. So the conical surface and the base which is only within this conical surface, the polygon itself, they form an object, a geometrical object which is called a pyramid. So that's the definition. The base for definition is the conical surface which we have explained in the prior lectures. Okay, now, so what are the elements of the pyramid? Now we are talking about basically terminology. Now these are point vertices, same thing as in the polygon, the points are called vertices. Now this is also a vertex, but this is a special vertex because it's actually an apex. So you can call it a vertex, but it's better to call it an apex than we will know what we are talking about. Reserving the word vertex for any of the points of the polygon which connect these segments to a figure. Okay, what else? So this is called a base. The polygon itself, it's a base. Okay, the a, s, b, s, c, s, d, s, etc. These are edges as well as a, b, b, c, c, d, d, e, etc. So all these segments which connect the points, the vertices of the pyramid including apex, all these segments are called edges. Now triangles, s, a, b, s, b, c, s, c, d, etc. These are all sides and the polygon a, b, c, d, f is the base. Now sides and base are commonly called faces of the pyramid. In particular the side faces sometimes are called lateral faces. Alright, so that's the terminology. Let me check if I have forgotten anything. Okay, that's it. Now the classification of the pyramid, well there are different pyramids obviously. For instance if you have a triangle as the base of the pyramid, and this actually must be dotted line because it's kind of invisible, right? So we have a triangle as the base and the pyramid is called, well guess what, triangular. Now you can have a square for instance, the base. Let me just draw another pyramid for you. So let's consider you have a square and I will view the square from the side, right? So that's why I will have something like this. And these are also invisible lines. So this is a square pyramid because at the base I have a square. That way Egyptian pyramids are square pyramids. Well obviously you can have a rectangle and then it will be a rectangular pyramid or a hexagonal and then it will be hexagonal pyramid etc. So you can differentiate different pyramids by the number of vertices of the base, 3, 4, 5, whatever the polygon is. Another important characteristic of the pyramid is its altitude or height. Well if you drop a perpendicular from the apex to the plane which contains the bases, this perpendicular is called basically an altitude or height of the pyramid. Now let me just draw a pyramid which might not actually be something which you imagine as a pyramid but nevertheless it's still a pyramid. So let me start from the same square in the beginning and then I will put my vertex over there. So it would be something like this. This is also a pyramid, not typical I would say, it's some kind of a skewed pyramid. And what would be the altitude of this pyramid? Well again if this is a plane where the base belongs to we have to drop a perpendicular to this plane and this perpendicular is an altitude or height of the pyramid. Now and the last kind of a pyramid which I would like to consider which probably would be something we would be dealing more often than others is consider you have a regular polygon as the base. Well let's say it's some kind of a hexagonal at the base and the apex has the perpendicular dropped to the plane falling exactly in the center of this regular polygon. Something like this. And this is the plane where the polygon belongs to. Well in this case if this is a regular polygon which means all sides are equal or angles are equal etc and this is the center of this regular polygon. And the apex projects down using the perpendicular to the plane, the base plane, if it projects exactly in the center it's kind of a regular pyramid with n sides. Sometimes it's called n pyramid. What's important is it's regular. Well now it's regular you obviously understand that all side edges are the same and all sides are also the same etc. These are all properties which we will be considering separately. Just as a terminology it's an n pyramid. Now what else is important in pyramid a specific case? What if I will have another plane, something like this and I will put this plane parallel to alpha so the beta plane is parallel to alpha plane and it intersects this perpendicular somewhere basically cutting the pyramid from the top. And that's what we will have as a result. So this plane beta is parallel to alpha and it cuts the top of the pyramid. Then it's called a truncated pyramid. Now by the way truncated pyramid you can always look at a United States currency bill $1 and you will have a picture of a truncated pyramid. It's some kind of a Masonic symbol. So truncated pyramid is also something which we will be dealing with as far as different problems are concerned. So that's probably it. It's all about pyramid I wanted to talk about, all these kind of terminology words which qualifies the elements of the pyramids. Now I would encourage you to look at notes to this lecture. I think it's important just to read it again. You will be more familiarized with the terminology because as soon as I will finish all these introductory lectures which basically bring you the concepts of solid geometry, then we will go through properties and I will just use this terminology very frequently. That's probably everything I wanted to talk about pyramids. Wait a minute, one more thing I think I forgot. The altitude of the truncated pyramid, the altitude of the truncated pyramid is obviously this piece of the perpendicular which we have used to have and we cut from the top this piece or whatever is left. Basically it's a distance between these two parallel planes. So any perpendicular between these two parallel planes would be equal to this particular altitude of the truncated pyramid. Now that's it. Thanks very much and good luck.