 Hi, I'm Zor. Welcome to Inezor Education. I would like to expand our menu of different prisms towards basically a general prism. General in terms of now we were talking about prism with rectangle or parallelogram as a base that was parallel to Pippit. We were talking about triangular prisms with triangle as a base. Well obviously we can talk about polygonal prisms when a polygon of different number of vertices is a directories of this prism. So just for example we can consider hexagonal prism. So that would be the picture of the hexagonal prism. Something like this. Looks like hexagonal prism. Hexagonal because there is a hexagon as a base or it can be pentagonal or quintagonal. Whatever it is. So most importantly is the following. The directories is a flat polygon of certain number of vertices. And all the side edges are obviously parallel to some line which basically makes the prism. Remember we had the concept of a cylindrical surface when there is one line and there is a directories, another line and we basically move this line in such a way that it slides through all the points of the directories. Making air and cylindrical surface. And then in case we have this directories as a polygon and then we cut another plane. So this is the base where the directories is and then somewhere else there is a parallel plane to the base. So they basically make the object called the prism. Alright. So what's important about polygonal prisms? Well first of all this isn't. So as usually I have the same formula for their volume which is area of the base times the altitude. Now how can I prove it? Actually this is much easier after we have already learned that the similar formula exists for triangular prism. Because what I can do is I can just divide it into triangular prisms like this. For instance this is A B C D E F. So I will connect one particular vertex with all others here and correspondingly here. So in this case I have cut my hexagonal prism into one, two, three, four triangular prism. One is A B C A prime B prime C prime. Another is A C D A prime C prime D prime. So these are vertical planes which cut my hexagonal prism into triangular. And obviously they all have the same altitude. Now their volume of the entire hexagonal prism is equal to some of these four triangular ones. And the area of the hexagon is equal to some of these areas of these bases. So basically you have like V1 is equal to S1 times H. V2 is equal to S2 times HV3 times H and V4 is equal to S4 times H. Where 1, 2, 3, 4 are 1, 2, 3, 4 triangular prisms. So if I will add them up I will have the total volume here and here I can factor out H and I will have a total base area times H. So the formula is still the same. What else is interesting about these polygonal prisms? Well there is others formula about the number of vertices, edges and faces. Now this formula is not only valid for prisms but for any kind of polyhedron. But in this case let's just check it out for the prism. So let's assume we have some kind of an n-sided polygon as a base. Now how many different let's say faces we have? Number of faces. Well if I have n sides of the polygon I have n side faces plus 2 bases so that gives me n plus 2. Now how many edges do I have? Well I have n edges here, n edges here and n side edges right? So it's 3n. And how many vertices we have? I have n vertices here and n vertices here. So it's 2n. And notice that f plus v minus e equals 2. And this is the Euler's formula. For any polyhedron, not only for prism but for any it can be whatever some non-necessarily prismatic kind of a polyhedron. The formula is true. So I'm not talking about any, I'm talking about the n-sided polygonal prism and for polygonal prism this formula is true as we see. Alright so that's basically the only interesting aspects of polygonal prisms, the volume and this correspondence between the number of faces, edges, and faces, edges, and vertices. So this is an interesting formula actually. If you will start checking this formula for any kind of object geometrical, just let me illustrate it for non-prismatic kind of object. Let's talk about pyramid for instance. If you have a triangular pyramid, so we have a triangle as a base and we have the top vertex. So how many vertices we have? Four, right? One, two, three, four. How many edges do we have? One, two, three, one, two, three, four, five, six. And how many faces do we have? One, two, three, and four. F four plus V four minus E six again two. So no matter what exactly you put as your figure with vertices, faces, and edges, the formula is true. We will check it again for any pyramid and obviously it will hold. Well basically that's it for today. That's all I wanted to talk about for polygonal prisms. Now as usually you know that this course is presented on Unizor.com and I usually recommend you to watch the lecture from this side because the site has comments and that's very important. Alright, that's it. Thank you very much and good luck.