 Hi, I'm Zor. Welcome to Unizor education. This lecture is part of the advanced mathematics course for teenagers presented on Unizor.com. Well, probably for older people as well if they're interested. I was kind of aiming for high school. Anyway, this is about volume of the pyramid. It's a very important topic because you cannot absolutely address this issue of the volume of a pyramid within the framework of the mathematics on the level of high school. You really need something like integration, etc. However, I have made a lot of preparations to make it as intuitively rigorous as possible. So I'm using the term intuitively rigorous. Yes, there are certain things which I left completely outside which need to be presented if you want to be an absolutely rigorous presenter. However, again, as I was saying, a lot of preparations were made to make it as rigorous as possible and intuitively obvious. So one of those preparations were related to Cavalieri principle. And this particular lecture will be about volume of the pyramid using the Cavalieri principle to evaluate this. The previous lecture was also dedicated to volume of the pyramid, but that was based on the limit theory. So if you remember, I was just slicing my pyramid into individual slices and built little prisms in every slice. That's a very valid approach, actually, which we'll be handling in some other cases. Now, this is just a different approach using this Cavalieri principle, as I was saying, which in turn actually also builds on the same kind of slicing thing. Whenever you would like to explain why this Cavalieri principle is actually correct, but again, accepting the principle as an axiom allows the shortcut, basically, all these logical conclusions. And that's exactly what I'm going to do. So let me first dedicate a few words to what actually Cavalieri principle is all about. Now, this is a very short description of a longer lecture in Cavalieri principle which is part of the previous topic related to 3D similarity. Sorry about that. Okay, so the Cavalieri principle was explained there, and I would definitely relate to that particular lecture for a complete description of it. Here it's just a very, very brief explanation what actually this Cavalieri principle is. So if you have two three-dimensional geometric figures, for instance, let's say you have a plane, you have a triangle here, and you have a triangle here. And let's assume that these triangles are of the same area. Now, let me just take a point somewhere and build a pyramid here and a pyramid here. Now, let's say I would like to draw another plane which is parallel to this one, and it actually does cutting between these two pyramids somehow, whatever. So if every plane parallel to this base plane cuts our two geometric figures, in this case pyramids, but doesn't really matter which one, and the sections are of the same area, then the Cavalieri principle says that the volume of these two geometric figures is exactly the same. And again, to rigorously prove it, you really have to go into the theory of limits, you have to slice the whole thing in little prisms, etc. So we are not talking about rigorous proof of this, we just take it as an axiom, if you wish. So this is some kind of a repetition of the Cavalieri principle. Now, obviously I assume that you know everything about pyramids as far as the terminology is concerned, so I will use whatever the terminology is available. Whatever is important is, yes, immediately after, in the previous topic, immediately after the Cavalieri principle, I gave a few theorems which are based on this Cavalieri principle, and they are very important actually. Now, one of these theorems, and I'm going to use it actually, is the following. Actually I referred to the previous picture which I was trying to draw. So if you have two pyramids, now I'm talking exactly about pyramids, not some abstract geometric figures, if you have one common apex with two pyramids, this is one, this is the plane where the base is lying, this is supposed to be like this, right? And you have another, this is the base, and you have the same point as apex. All right, I think these two are supposed to be something like this. Now, if these two triangles are of the same area, then the pyramids have the same volume. This is a very easy theorem which I have proved in this lecture dedicated to consequences from Cavalieri principle. So these two triangles are in the same plane, they have the same area, and they have the same, and pyramids have the same apex. All right, so another theorem was basically immediate consequence of this one, if instead of triangles you have parallelogram here, and you divide it by diagonal into triangles, and you have a pyramid like this. Then our quadrilateral pyramid is divided by this diagonal and these two edges in two halves. And these two halves have the same volume. And this is immediate consequence from the previous because obviously these two triangles in the base have the same area because it's a parallelogram and it's diagonal, so divides the parallelogram into two congruent triangles. Okay, so this is another theorem which I'm going to use. So this is just all introduction into whatever will be presented today. This is something which you have to have as your knowledge, as something which you know about and you can use it. Now I would refer to previous lecture again where we have concluded using the limit theory this formula that the volume of the pyramid is one third of area of the base times altitude. Now this was done in the previous lecture and the methodology was slicing the pyramid into thin slices. Each one of them is presented as approximated as a prism. Now I'm going to try to derive the same formula using the Cavalier principle. Now the proof itself is extremely simple. So all whatever explanations which I'm basically doing right now is just to basically approach this particular very simple logic. But again it's simple because you have to know everything I'm talking about before that. So that's why I'm repeating all these things. So right now what I'm going to do is the following. I have a pyramid, let's say I have a pyramid something like this. And again my purpose is to derive this formula. What I'm going to do is I will make a construction which will expand this pyramid to a prism with the same base so it will be a triangular prism and the same height. So to make it the same height what I will do is the following. Let's say these are vertices. Now this is the plane beta where my base triangle ABC is lying. And Apex S is outside. So through S I will construct another plane which is parallel to my plane beta called gamma. So I'm positioning my gamma as parallel to the beta and going through this point S. Now what does it mean? In particular it means that the distance between these two planes is exactly the same as the altitude of the pyramid. Because altitude is perpendicular to the base. So it's perpendicular to another plane which is parallel to the base plane. And perpendicular to both planes parallel to each other is the distance between these planes. So the distance between these planes is exactly as altitude. Now what I will do next is from B and C I will draw lines parallel to SA. That's how I will convert my pyramid into prism so it would be something like this. This would be B prime and C would be something like here. C prime. So now we're talking about prism which has a base ABC. This is the bottom base and the top base SB prime C prime. So parallel to SA and B prime and C prime balloon to this plane gamma. So the height of this prism is exactly the same as the height of the pyramid. Now I have to change this into invisible because it's on the back. Now next what I will do I will connect B prime and C with a diagonal. Now you need a little bit of imagination because basically this is it. I have almost finished proving my formula because what I would like to say is that my prism, the area of the base is exactly the same as the area of the pyramid. The altitude of the prism is exactly as the altitude of the pyramid. So I know that the volume of prism is equal to s times h, right? Now I should use lowercase s here. So it doesn't mix with my x letter. So lowercase s is area of triangle ABC and lowercase h is length of sh, right? So I know that this is the volume of my prism. We have actually covered the volume of the prism in previous lectures. So what I would like to show right now that this prism can be divided into three equal parts and our pyramid S ABC is one of those three parts and I will tell you the other two. And since they're all having the same volume, the volume of each of them is one third. Okay, let's talk about this. Okay, first of all we have to just imagine which three prisms, pyramids I'm talking about. One pyramid has apex at C and base S B prime C prime. So it's on this side. Another is our original one, apex at S and ABC is its base. It's on that side from the left. And in between I have another pyramid which I can use for instance apex as C and base S B B prime. That would be another third pyramid. So these three pyramids combined together make up our prism. So again, let's imagine ourselves. The prism from that side which I can use apex C and base S B prime C prime. The original prism which is on the left side, on the left and the back actually, which is S ABC. And then squeezed in the middle is the pyramid which I can use C as a apex and S B B prime as a base. But I would like to show that the volume of all three of them is the same. Okay, let's start with left and right. So the left one is original S ABC and from the right I have C as apex and S B prime C prime as a base. Now, they're talking about prism, right? So the bases are the same. So the S B prime C prime is exactly the same as ABC. Except this pyramid is upside down. It has exactly the same base and it has exactly the same height, obviously, because the height is again the distance between these two planes, which is the same for these two pyramids. It's distance from C to this plane or from S to this plane, right? So the heights are the same and the bases are basically, well, they're congruent because it's a prism. So what I can do is let me just turn this pyramid, this pyramid upside down and position S B prime C prime somewhere on this plane in such a way that point C would coincide with point S, right? I can do it. Whatever the position of the base will be, I don't care as long as the base is on the plane beta. And the C is replaced with S. Now, back to the theorem, which I was talking about before. You have on one plane two triangles, which are congruent to each other. Therefore, they have the same area and they share the same apex. Now, based on this theorem, which I quoted before and I proved in the previous topic, I think the lecture was called mini theorems. It was a theorem B. The volume of these two is the same because they have the same area of the bases and they have their apexes at the same position, which means they have the same altitude. So, fine. So these are the same. Now, the next thing is, I would like to say that this pyramid which is in between these two has the same volume as original S A B C. Now, the way how I will do it is the following. Now, what is a triangular pyramid if you consider it from different aspects? Now, this is my original, but this is a triangular pyramid, which means I can actually take any other point, call it apex and the rest I will call a base. It's exactly the same pyramid. In the case of triangular pyramid, any vertex can serve as apex and the other three would be considered a triangle, which is a base, right? So, in this particular case, my original pyramid I will consider not as S as an apex and A B C as a base, but C as an apex and S A B as a base. Now, my third pyramid is C as an apex and S B B prime as a base. So, let's consider these two pyramids. They share the same apex C and the bases are S A B and S B B prime. Now, this is parallelogram. Remember, we're talking about prism, so every side edge is, every side face is a parallelogram and this is the diagonal of the parallelogram, which means it divides into congruent triangles. So, we have exactly the situation, which I was quoting before as yet another theorem, when you have a parallelogram on the plane divided in two halves and you have some kind of an apex and you consider one pyramid against another pyramid and they have the same volume. Because, again, we have the same bases as far as the volume is concerned and they share the common apex, which means it's the same height. So, this middle pyramid, which is S as an apex and S B D prime as a base has exactly the same volume as our original pyramid, S A B C. Consider it not as S A B C, but C S A B, which doesn't really matter, right? So, that's the end of it. We have proven that the volume of the right and the left one is the same and the volume of the middle one and the left one is the same. So, everything, all three pyramids have the same volume, which means that each one of them has one third of this volume. And this is the end of it. So, the proof itself was relatively short. It's more of a talking whatever was leading to this particular proof. So, all the preparations were made and I was trying to make these preparations as intuitively obvious as possible. So, the very proof is relatively short and simple. So, all you have to do is to prove that these three pyramids have the same volume and I based this on the theorems which I have proven before and those are based on Cavalier principle. So, that's how Cavalier principle acts if you would like to prove this formula. Without resorting to limit theory etc. But obviously, you have to understand that limit theory is in the heart of the Cavalier principle anyway if you would like to prove it instead of taking it as an axiom. Well, that's it for today and I recommend you to read the notes to this lecture on Unizor.com. Notes might be written in more details than I am just explaining maybe I forgot something, but I don't think I forgot. I think everything is fine. But notes are still very, very important to read so I do recommend you to go through them. Thanks very much and good luck.