 Hi, I'm Zor. Welcome to Unizor education. This is a continuation of a solid geometry part of advanced mathematics course for teenagers presented on Unizor.com We have spent a considerable amount of time talking about lines and planes and now we will combine these elements and our analytical knowledge into something more complex So today we will talk about prisms and in particular about parallel epipids All right, so first of all, let me just remind you Something about prisms what actually called prisms we start from cylindrical surface cylindrical surface is formed if you have some kind of a line in space Not necessarily a plane line. It can be in space real and you have some Straight line somewhere else now this line is called this curve is called directors and this one is called Generatrix and now consider a surface which is formed by all the lines parallel to this one Going through all the points of this curve Something like this So that would be some kind of a surface which we generally call a cylindrical surface Now the prism is a particular cylindrical surface and a combination of cylindrical surface and a couple of planes which we call base So let's imagine that this particular directors Is not just any curve in space But it's contained On some kind of a plane and this is a parallelogram So this is the plane this is a parallelogram on that plane and this is our director So we will draw lines parallel to this one To make a cylindrical surface and then we will have well this actually supports our Cylindrical surface from below and now we will have another plane parallel to this one which cuts it as well along some Well, we will prove it's a parallel so This figure which is formed by cylindrical surface as sides and two parallelograms as the basis the upper and and and the lower base This is called the prism and we did talk about this before So we need cylindrical surface with directors being a parallelogram on some kind of a plane and Another plane which is parallel to this one. So it's two base planes now This figure Actually can be general generally called Hexach had wrong because there are six Flat surfaces one two and four on the side So that's six. That's why hexach had wrong Wow, what's necessary to do right now is to basically Prove the very important and the first probably property of any Prism of that side of that kind That all surfaces all faces are actual parallelograms in this case and By the way, I Will sell it I will say say before I will prove it But actually this is a statement which should be said after the proof of this theorem You see this particular type of prism where the The base is parallelogram that directors is a parallelogram and it's basically restricted from Top and bottom by these two parallelogram. It's called parallel epipod. However, there is a very important part of this Word parallel epipod. It's parallel now. It's parallelograms which are on each side or for side sides and both Basis and that's exactly what we're going to prove right now which justifies the name parallel epipod of this particular prism so again the prism Contains a cylindrical surface with the directors of a parallelogram. So these lines are all parallel to this one and Another plane parallel to this one Which restricts from from the top? So the first theory in which I'm going to prove about this figure which we can Which we will have the right to call parallel epipod is that every side every face of this Figure is a parallelogram Well, let's start from the bottom bottom is parallelogram by definition Because that's exactly how we started we said directors is supposed to be a parallelogram lying in some base plane Okay, that's done Now all these lines Are parallel to To the generator right so which means they are parallel to themselves so these lines are all parallel Now the plane is parallel to this plane Now this is a line right So This point this this and this The vertices of this figure Now since these two lines are parallel They are Lying in the same plane and any other line connecting these two Edges this and this also lies within the same plane because if it's not then I can always take the point and Within this plane defined by these four points. I can draw a parallel line to this one and I will have two different lines parallel to this one Containing the same point one within the plane and One outside of the plane so there is no such thing. So all all the lines which are Connecting these points. They are all within the same plane. So this is really a plane So it's not just these four points lie on the same plane because these lines are parallel The whole Face is lying in the same plane Same thing with all other four sides All five four side faces of this figure each of them is a plane now It's a plane Now this line is an intersection of this plane and This plane the top base but the top base is parallel to the bottom base and we have Proved a theorem that if you have two planes parallel to each other and The plane which intersects them both then the intersection lines are parallel to themselves so these two lines are parallel and these two lines are parallel which proves that this is parallelogram and That's for each other side Now what remains to be proven is that the top is parallel Parallelogram as well, but that's easy because this is parallel to this Now this is parallel to that but these are in between themselves parallel. So these are supposed to be parallel They're all parallel To each other Similarly these two lines. They're parallel to these two these two are parallel to themselves. So these are parallel so on the top we also have a Quadrilateral with opposite sides parallel to each other, which is parallelogram. So that proves that every face side faces Bottom base and the top base all of them are parallelogram So parallelepiped is a justified name now in many cases I've seen that they define parallel epiped as a figure with six Sides six faces if you wish Each of them is parallelogram Well, I don't like this definition because it's really a theory in which we can prove if we take a much simpler definition Which we start with a prism which is a cylindrical surface and then define the directors Because if you are just saying that this is a figure with six Parallelograms as as faces you really have to prove its existence. Maybe they don't exist Now this particular definition which is based on the cylindrical surface. It's a constructive way of building a Parallel epiped and that's why I preferred now Let's just forget about this definition and let's concentrate on parallel epiped Itself So we have Parallel epiped which can be drawn Something like this. This is invisible. Okay That's nice program even for my artistic talents Okay, so a couple of definitions about this parallelogram So every flat surface is called a face These four are side faces this one back one and the left These are base faces if you wish top and bottom. I mean doesn't really matter how you call it Now these are edges These are all edges Now the points are vertices but each of them are vertex So we have six different faces and that's why it's hexahedron. We have What four and four and four we have 12 edges and We have eight eight vertices Now we also have diagonals Diagonal can be side like this one for instance These are two side diagonals Diagonal can be base on this base for instance. I have these two diagonals and on that base. I have these two diagonals And Diagonals can be space diagonals, which means they are inside the parallel epiped. Let's say diagonal from B prime To D from further top Vertex to a Nearer bottom or from A to C prime Or from A prime to C. These are all space diagonals so that's Not much for elements of this parallel epiped Now there is a concept of a right parallel epiped. Now remember what the right angle is, right? It's when the perpendicular thing now the right Purpin parallel epip parallel epiped is when Every side edge or if you wish Generatrix they're all parallel to Is perpendicular to the plane of the Where the base is? So let's go back to the construction of the parallel epiped Remember there is a base plane in which we have parallelogram and then that's a directors and then there is a generatrix And now we built all the sides basically the cylindrical surface From line parallel to this one. So if this line is perpendicular to The base then every edge would be Obviously perpendicular to this plane to the base plane well and to this base plane because they're parallel So the right Parallel epiped is the one when the side edges are perpendicular to bases Sorry So that's the right Parallel epiped now So right parallel epiped is just one particular kind of parallel epiped Which we can go general and the only characteristic of the right parallel epiped is perpendicularity of the side edges Nothing about base now. Let's add another Requirements for the base what if base not just plain parallelogram, but a rectangle so number one we have a rectangle in the base and We have perpendicularity of the edges side edges to the base then This particular parallel epiped is called rectangular Parallel epiped and to tell the truth in most cases We probably will be dealing with rectangular parallel epiped Which means you have a rectangle in the base and edges are perpendicular to the base That's kind of a simplest form and it has lots of good properties Okay, now We can even further restrict rectangular Parallel epipeds what if this is not just a rectangle at the base But a square so all sides are equal and what's important as well the edges Also have the length equal to this length of this side of the square Then it's a cube, right? So we have All faces are Squares of the same size the same side with the same lengths of the of the edge So this is equal to this is equal to this these are all perpendicular Because this edge is perpendicular to the plane which means it's perpendicular to this line and to this line So they're all squares of the same size Then it's a cube. Oh, by the way, I forgot actually rectangular Parallel epiped sometimes are called cuboids Well, I rarely use this word that I would prefer rectangular parallel epiped. Well because it's actually Like a cube because it's kind of a straight edges, etc. But it's not the cube because the the side Edge is not equal to the bottom edge Then it's a cube, but if it's not equal, they're saying it's cuboid, but anyway, I prefer rectangular Parallel epiped all right Now what's important about? general parallel epiped General I mean there is no requirement about Right angles, etc. Well the opposite sides this and this or this and this or the front and the back all these opposite sides are basically equal Parallel epipeds, sorry parallelograms parallelograms Well the fact that these are parallelograms we have already proven right, but now if you think about this Let's say the top and the bottom Now this side Because this is parallelogram equal to this one and This side is equal to correspondingly the BC now a prime B prime is equal to a B. So these are two parallelograms with Exactly equal Sides now they are equal in lengths and They are parallel, which means angles are also the same since this is parallel to this this is parallel to this So the angle B prime a prime D prime is the same as BAG, right? so we have two parallel two parallelograms with equal sides and Equal angles so they are Equal to each other or Congrant to each other if you wish so opposite faces Top and bottom left and right front and back are Exactly congruent to each other Now we will prove another interesting theorem. I have defined Space diagonals remember from like B prime to D. Let's forget about other diagonals So they don't interfere So we're talking only about space diagonals and These guys too. Okay. We have space diagonal from B prime to D and Let's take another space diagonal. Let's say from C prime To a and we can take any other pair So now the theorem is all space diagonals Intersect each other In one and the same point and this point divides each diagonal in half now Actually, it's a very simple theory because consider B prime C prime G and a You see B prime C prime is parallel to BC BC's parallel to a D So B prime C prime is parallel to a G. So these two lines are parallel and Obviously, they're equal in length because this is equal to this and this is equal to that which means that a B prime C prime G is parallelogram Because opposite sides are parallel and equal in lengths But in the parallelogram, we know the diagonals and these are diagonals of this parallelogram So let me just draw this parallelogram. So this is the line and this is the line so this Parallelogram just cuts this particular Parallel epipids in two diagonally from the back The edge on the top to the Front edge on the bottom Or we can actually take any other pair doesn't really matter the same exactly thing. So any diagonals any two space diagonals are actually diagonals in the Parallelogram in this case B prime C prime g8 and Again, we know about the diagonals of the parallelogram that they are intersecting in the middle so this diagonal is Intersecting with this and this is the middle point, but we can take some other pair. Let's say a prime C now by By using a different Parallelogram a prime B prime CD we see that this B prime G is intersecting with a prime C also in the middle of both of them. So it's exactly the same point. It's a middle of B prime D So every other diagonal would intersect any other diagonal right in the middle So it's supposed to be one and only point And this point is the middle of every space diagonal Now Now we will consider something which I would qualify as three-dimensional equivalent of the Pythagorean theorem So let's consider we have a Rectangular Parallelipid this is my front side This is my top and Obviously I can draw these two now It's rectangular Parallelipid which means Every side is a rectangle So let's call this A this B and this C and A B C D A B C And I will also put The space diagonal I will call it D Well, let me remind you the corresponding theorem on the on the plane if you have A and B rectangle Rectangle and this is diagonal C then C square is equal to a square plus B square, right? This is a theorem Pythagorean theorem, right? Now the equivalent in three-dimensional case is D square is equal to a square plus B square plus C square So the square of a diagonal of a rectangular it's very important rectangular Parallelipid is equal to sum of squares of three lines three ages which share the same vertex with this space diagonal now Proof is trivial quite frankly, so let's connect a to C now, obviously a C C prime is Right triangle so a C square plus C C square equals to a C prime square a C square plus C C prime Square is equal to a C prime square from the right triangle a C C prime now a C square in turn we can replace with a D square plus C D square Right So basically I have a D square, which is a square plus C D square C D is the same as a B, which is B square plus C C square Which is the same as a a square a prime, which is C square equals to my D square Which is a C square. So that's the proof very easy and the last which I wanted to point out to is What is the area of all the sides? ages all the side sides and and two bases together, what's the area of this particular? Rectangular parallelepipid Well, let's just think about first of all you remember that we were talking about the opposite Faces are exactly the same congruent to each other. So this is a rectangular Parallelipipid. So what's the area of the base? Well, it's a times B, right? But we have two bases, so it's two AB now the front Front is a times C But we have a back which is exactly the same. So it's two AC and now left and right now this is B and C and again two of them so this is the total area of the surface of all the Faces of rectangular Parallelipipid Or you can obviously factor out to two times AB plus AC plus BC. Well, basically, that's all I wanted to talk about Parallelipipids. What's very important is the volume Volume is a very interesting thing and I will dedicate the whole lecture to volumes of Parallelipipids. That would be my next lecture and meanwhile that's it for today. I think I've done everything. Thank you very much and good luck