 Hi, I'm Zor. Welcome to Unizor education. I would like to introduce you something which is called a Cavalieri principle. Cavalieri is mathematician. Now, this lecture is part of the advanced mathematics course for high school students. It's presented on Unizor.com. That's where I suggest you to watch this lecture. And the Cavalieri principle I basically would like to use it to evaluate the volume of the pyramid. That's probably the most important goal which I put for this principle. So, this lecture is basically explaining what this principle is about. I put this lecture as part of the 3D similarity topic. Primarily because for the purpose which I am going to use this principle, which is the volume of the pyramid, I will use both the Cavalieri principle and the 3D three-dimensional similarity. So, all right, so let's just talk about this principle. Interesting point is about measure in general in mathematics. It's not such a simple thing. I mean, it's not like you have let's say square and you measure its area by multiplying widths by lengths. You see, in geometry we have many different non-quadratic type shapes. So, it's easy to do it if this is a square one by one, then we just kind of postulate that the area is equal to one. But then if the area is some irrational number lengths and irrational number widths, how can we measure in these units? So, we actually need the whole apparatus of higher level mathematics which involves integration limit theory obviously and integration to approach this rigorously. But I would still like to approach it for the level of significantly below that higher level of mathematics, the level of the high school students. So, that's why we're trying to simplify this and make some shortcuts. So, the presentation is not 100% rigorous in a rigorousness which mathematics is used to have. But still, I will try to put some explanation and the foundation for this limit based in limit theory, the foundation for the Cavalier principle based on the limit theory, which would explain it in a reasonably intuitive kind of form. All right. So, there are two kind of different types of Cavalier principle. One is two-dimensional and other is three-dimensional. Again, my purpose is to measure the volume of the pyramid which is three-dimensional case. Two-dimensional seems to be much simpler and that's why I would probably start with explanation of what is in two-dimensional case and then we will go to three-dimensional. So, in two-dimensional case, all right, let me just state this principle. Let's say you have two objects, this one and this one. And let's assume that there is some kind of a baseline and let's further assume that any line which is parallel to the baseline has intersection with x equally measured with intersection with y. Let's just assume that these two, let's call it A, B, C, D. So, let's assume that segments are the same. For any line H which is parallel to the baseline D, this should be true. So, A prime, B prime is equal to C prime, D prime. For any line, so if this assumption is true, then the Cavalier principle states that the area of these two is exactly the same. So, if any line parallel to the base intersects in one object along the same segment as in another, then the areas are the same. Now, by the way, sometimes it might not be just one segment, it can be two segments, something like this. Then intersection would be a combination of two segments, this plus this on this side and this plus that on that side. So, if some of these is equal to some of them, that actually is the same thing. So, if any line intersects along the segment or a group of segments which together equal in lengths to the intersection with another object, then the areas are the same. Now, this is a principle which we can take without the proof and basically make a shortcut because in theory, in the real mathematics, this can be proven. But it's not easy to do it rigorously. So, it involves the calculus of integration and stuff like this which we did not really touch yet. But in any case, as a shortcut, it's probably good enough for us. Now, I will explain the foundation to this principle in a non-rigorous, in a more intuitive fashion. It's not like it is explained in the higher levels of mathematics. But I think I will present it intuitively enough for everybody to believe that this principle is not just off the wall. It really has some very solid foundation. Now, if we use this principle. Now, what happens if we use this principle? Well, for instance, using this principle, it's very easy to derive the formula of the area of parallelogram. For instance, now, we all know what is the area of the rectangle, right? It's A times B. Now, what if we have a parallelogram which has exactly the same base A, but it looks like this? Now, the altitude is exactly the same B. But is the area of this equal to area of this? Well, yes. I know you can always say that geometrically, we can prove it by drawing these two perpendiculars, then proving that this triangle is congruent to this one. And that's why if we will cut this one and add to this part, it will be the figure of the same area. And now, it's the same A times B, which is base times altitude. That's true. However, if we don't do this, we can use the principle of Cavalieri because any line parallel to the base will cut this is equal to A and this is equal to A because it's the same parallelogram, right? So this is also A. So every section will have the same length here and there. So from the Cavalieri principle, it basically follows immediately. You don't have to construct triangles. You don't have to prove that they are equal, etc., etc. So Cavalieri principle allows us to shortcut certain proofs. Now, in other cases, like, for instance, in the case of a three-dimensional pyramid, and I will talk about three-dimensional principle, Cavalieri, it actually makes it much, much easier. All right. Now, so the principle of Cavalieri, if it's true, if it's actually held in this particular case, for instance, then from this principle follows the equality of the areas. How about the other way around? If you have two different figures of the same areas, does it mean that I can cut them with certain section and it will be the same length? Absolutely not. So this principle is sufficient for equality of the areas, but definitely it's not necessary. An obvious example is let's take a square 2 by 2 and rectangle 1 by 4. Area is the same, but obviously there is no such section, base, whatever, which will give you every line which is parallel. We would give you exactly the same length of this segment as that segment. So again, Cavalieri principle is a sufficient condition for equality of the areas in this case or volumes in the three-dimensional case. It's not necessary. All right, what else is necessary? Okay, now it's time to explain what this is in a three-dimensional case. Actually, it's very similar. Let us imagine that we have two objects and I will use right rectangular prism and another prism which has exactly the same base, but not the right one. No, that's not good enough. Something like this. So let's assume that both have the same base and the same altitude, but this one is right, which means these angles are perpendicular and this one is slanted. Now, if I will draw a plane parallel to the base, it will cut something like this from here and something like this from here. Now, both cuts will be equal to the base, right? It will be the same rectangle in this case, right? And rectangle in this case, and they are equal to each other, which means every plane will cut it across exactly the same kind of a polygon and obviously they have the same area. So the principle of Cavalieri states the following. If you have two objects in space, in three-dimensional space and some kind of a base plane, well, in this case my base plane is this one, where both of them are standing, base plane. And if any plane which cuts both of these two objects cuts along the figure which has exactly the same area with one as with another, then the volumes are equal. And in this case, for instance, this is true. Now, even if I will change this rectangle to a parallelogram, which has exactly the same this side and altitude as this one. So if this is A, this is B, and this is a rectangle at the base, now this is A and this is B, which is altitude. So this is just slanted parallel, slanted rectangle. So it's a parallelogram. And I also slant the altitude. It will still be exactly the same volume. Why? Because this parallelogram has exactly the same area as this one, because it has the same altitude as this one and the same base. So the area of each section would be exactly the same parallelogram. And since these areas are equal, these areas are equal. So every plane cuts it along the polygon of the same area. Then the volumes will be the same. So again, this is a principle, which means we will just use it without the proof. However, I will provide a relatively, well, intuitively obvious, let's say, way to approach this, to understand it thoroughly. All right? So that's basically what the principle is. Now, I will not use it right now in this lecture to derive the volume of the pyramid. What I would like actually to talk about is about this foundation, this intuitive foundation, which is using the limit theory, at least in one case, and then maybe in two-dimensional, maybe in three-dimensional as well, if we have some time. All right. So, yeah, by the way, one more thing. Obviously, as I was saying in the two-dimensional case, this principle is sufficient for equality of the volume, but it's not really necessary because, again, you can have something like 2 by 2 by 2 cube and 1 by 2 by 4 prism. They have the same volume, which is the multiplication of these, which is 8 and 8, but they will never be in such a position that I can cut old planes parallel to the base plane, which will cut the same area, polygons. All right. So, sufficient, but not necessary. So now, I would like to spend some time to explain the foundation of this principle. How Cavalieri basically came to this principle without, you know, using a very rigorous considerations of contemporary mathematics, but it was rigorous enough, I'm sure, for his understanding. So, let's go into two-dimensional case, and let's say we will have two triangles, for instance, one and the same base, but I will have it something like this. Now, this is the base, the base line. Now, it's parallel to this one. Now, what I will do is the following. I will, the distance between these two parallel lines, this is my altitude to both of them, it's called h, I will divide it into n equal parts and draw parallel lines. Now, each parallel line cuts the same segment on both sides, this equal to this, this equal to this, this equal to this. Right? So, let's call this a 1, a 2, a 3, b 1, b 2, b 3, b 1, b 2, b 3, e 1, e 2, e 3. Now, so far, the Cavalieri principle tells me that a 1 b 1 equals to d 1 e 1, etc. So, what I will do next is I will draw perpendicular, I will cut my triangle into a set of rectangles, something like this. Now, the altitudes of each of these is the same because it's the distance between two parallel lines. Now, the base of each rectangle, this equals to this, this equals to this, this equals to this, is also the same. So, each of these rectangles, each pair of these rectangles are congruent. So, they have the same area. And now, I will use the limit theory in the following way. Now, this is, I divide it into 1, 2, 3, 4 parts. Now, the more divisions I have, let's say I will divide it into 100 parts, then the stack of these rectangles will more and more resemble the triangle and this one. So, for each case, no matter how many times, how many segments I break this altitude into, the sum of these rectangles will be equal to sum of these rectangles. And since with infinite increasing, infinitely large number of divisions, I will get infinitely close to the triangle, this one and this one. And since on each step, I will have a quality between areas of these stacked rectangles, one on the top of another. And in both cases, this stack will approximate this triangle and this stack will approximate that triangle, then it's reasonable to assume that in the limit, when my number is infinite, the principle will basically hold that the areas will be the same. Since on every step it's the same, it's reasonable to assume it will be at the end the same. Now, it's not a rigorous proof. It's a very, I would say, intuitively understandable explanation. Now, in the three-dimensional case, it's similar. So, let's say we have two slanted prisms. Let's say this one and this one, something like this. But the basis, let's just assume the basis have the same area, all right? And since these are prisms, then every plane parallel to the base would cut the area similar to this one. And that will do exactly the same kind of considerations. So, if I will have a cut, something like this, one, and I will divide the altitude in a certain number of parts. So, the next plane would be something like this. And I will draw a right prism. Now, it will be the right prism in this case. By dropping perpendicular from one plane to another. So, I will basically convert my initial shape, whatever the shape was. I mean, not necessarily actually the prism, whatever the shape was. Into a stack of prisms, into a stack of right prisms, which volume we know how to calculate. It's the area times the altitude. And same thing will be here. And this area will be much close, infinitely close to this one. And this one will be infinitely close to this one. These are prisms. But in any case, if the principle of Cavalieri is held, I don't even care about the basis. I care that each plane which cuts both figures will cut the same area. And if it's the same area, it will be the same volume of this little prism, because I'm dropping the perpendicular of the same height in this case. So, the altitude of these prisms will be the same. So, any polyhedron I have here and there, it will cut certain polygon. And if these two polygons are of the same area, then these little prisms will be of the same volume. And now I'm just stacking one on the top of another. And each one of them is exactly equal, left equal to right. And then I will increase the number of divisions. So, very briefly, that's the same kind of a logic. The more I, the more number of little segments which I'm cutting, the altitude, the closer this stack of prisms resemble my polyhedron. Now, even in the more complicated case, when it's not a polyhedron, maybe it's just a sphere or a cone or something like that, it's still the same kind of approximation. The only thing is we might actually approximate a circle with a polygon and then a cone with a stack of these prisms, polygonal prisms. So, that's basically the kind of approach, which I wanted to point you to as a direction of the proof. It's not exactly the proof, but nevertheless it is very close to. Now, well, basically that's all I wanted to talk about today. This is just an introduction into the Cavalier principle. I want you to understand that this principle is not an axiom which we should really take without the proof. The Rieser proof, we just skip the proof to shortcut the something like volume of the pyramid, which will be in the lecture. I will probably present a couple of problems as well for this principle, but generally speaking, that's all I wanted to talk about today. I would like you to read the notes for this lecture. It's on Unisor.com. It's very helpful. And other than that, thanks very much and good luck.