 Hi, I'm Zor. Welcome to Unisor Education. We continue the course of advanced mathematics for high school students. It's presented on Unisor.com. That's where I suggest you to watch this lecture from, because there are notes for every, including this one lecture. Also, there is some additional functionality on this Unisor.com website, which basically helps you to organize the whole educational process, including many problems solving and exams. Alright, so right now our topic is the cones, its area and volume of cones. Alright, so first of all, let's just very briefly recall what the cone is and what exactly we will be dealing with, what kind of cones we will be dealing with. Well, first of all, I have introduced a concept of a conical surface, which includes some fixed point and some curve in space, which is called directories. And now every point on this curve is connected with this point with a straight line, thereby forming some kind of a surface actually goes both ways. Now, when I'm talking about cones in this particular lecture and in most of the problems, what I really mean is, number one, this curve is actually a circle, a flat circle, which means there is some kind of a plane on which the circle is located. And also this particular point through which all the lines are going through, it's projecting at the center of this cone. And now, again when I'm talking about cone, I mean only the part which is from this point to the circle. And this is basically the cone which we will be dealing with. It's called the right cone because this perpendicular to the center is the projection of the apex, this is called apex, and this is base, obviously. So that's what we will be dealing with. A particular kind of cones, which is basically the right cone, and when we are talking about area and volume, we are talking about area and volume of this particular geometric object. Area would be consisting from the area of the side of the cone and the area of the base, the circle, and the volume is basically everything within this particular geometric object. Okay, so that's very brief introduction into what exactly our subject of investigation in this lecture. Right cones, okay. Now, what kind of parameters define the cone? Well, actually there are only two parameters. And usually the parameters we will be talking to are the radius of the cone and its altitude, h. Now, radius defines a circle. Now, the height defines the apex because we have to take from the center perpendicular, and this is the length of this perpendicular. So it's completely defines both a circle and the position of the apex above this circle, and that's why it defines completely the surface, the conical surface, and the entire cone. So r and h, radius and the altitude or height of the cone are two parameters we will be dealing with. Well, there is another important characteristic which is the lengths of this particular segment from the apex. To the base, let's call it l. Now, obviously from the Pythagorean theorem, l is equal to the square root of r squared plus h squared, because since this is a projection from the apex, it's perpendicular to the circle and it goes into the center. That's why this is the right angle. So every point on the circle connected with an apex would be a hypotenuse of the corresponding triangle, right triangle. This one, this one, all these, this one, this one, S-A-O, S-B-O, S-C-O, S-G-O, they're all the same type of triangles, they're all congruent to each other. They have common altitude, which is one catheter, and another catheter is the radius of a circle, which is also the same for all of these points, a, b, c, and g. That's why hypotenuses are the same. If you look at the cone along the line which connects the apex with every point on the base circle, it will be a segment of the same length, and this is the length, completely defined by two main characteristics, r and h. Now, in terms of these r and h, we would like to express our surface area of the cone and the value. Alright, so, surface area of the cone. Well, let's first of all talk about the side surface, the surface of the conical surface, the area of the conical surface, because obviously we can always add the area of the base pi r square without any problems. So, the area of the side surface. Well, have you ever made a conical hat? You know, people are wearing, like in Halloween, for instance, all these witches, they're wearing these conical hats. Now, if you have ever been able to do it from the paper, for instance, what kind of paper would you take for this? Well, if you think it's a triangle and you just wrap it around and glue along these two lines, that would not be the proper conical hat. Why? Well, because this is shorter, this is even shortest, this is the longer line. So, you will get a hat with one side being a little bit longer than another, so it will be a cone like this. So, it would not be a right cone. This point would not project down. To make the right cone, you have to make all these segments of the same length, because they're all supposed to be these sides along the side surface area, which means this should be a circle. This is the proper beginning of the conical hat. Now, if you will put them together and glue along these lines, you will have a perfect circle at the base, and the top will project right into the center of this base, like in this particular case. So, back to the area. Let's just cut our cone along one of the side segments, like SA, for instance, and open it up. Now, these are all straight lines. So, I'm not proving this rigorously, but you obviously feel that you will cut the cone here and roll it out on the flat surface, you will get something like this. Maybe a little bit bigger radius, something like this. Now, what would happen? The length of this circle would be this, right? Because I will open it up. So, that would be my length of my circle. Let's say this is A, this is B, this is C, this is B, for instance, something like this. So, if we cut it open and roll it out on the flat surface, we will get something like this. And what would be the length or the radius of this circle, if you wish? Well, the radius of this circle will be the length of this side segment, which is this. L. So, I know the L, the radius of this circle, and the area of the side surface of the cone will be the area of this sector. This is a sector of a full circle, right? You can imagine that there is some kind of a circle and this is just a sector within this big circle. Now, how can we calculate the area of this sector? I know the radius, which is L. I know the whole circle, which means I know its circumference and I know its area. Now, I know this length. So, if this length, if I will divide this length by circumference of the whole circle, I will have the piece of the big circle, which this area actually takes. So, this area, let's call it area of a sector. If you divide it to the area of a circle, it would be the same as this length of the sector divided by circumference of a circle. So, again, this length of this arc divided by the total circumference of this circle is equal to this area divided by the total area. That's obvious, right? Okay. So, the area of the sector, we don't know. Let's call it x divided by the area of the circle. Now, circle has the radius L, so it's pi L squared equals to this length. Now, what is this length? Remember, this is the circumference of this circle of radius r. So, it's 2 pi r divided by circle circumference, which is radius L, so it's 2 pi L. So, what do I get from this? Pi and pi, L and L, 2 and 2, and I have x equals pi L r, pi L r. So, that's my answer. That's my area of the side of the cone. So, we know what L is, so I have that area of the side equals pi L r. So, pi r and square root of r squared plus h squared. Now, if I will add to this the area of the base, I will get a complete area of a cone, which is equal to pi r squared, the area of the circle, plus pi r squared root of r squared plus h squared, which is equal to pi r r plus square root of r squared plus h squared. Now, that's basically the formula, which I would quite frankly never remember this formula, but it's relatively trivial to derive it from whatever the consideration we just made. Just cut the cone and roll it out. All you have to do is just understand the proportionality between the sector and the total circle. So, is it a rigorous proof of the formula? Well, quite frankly, not exactly. To be completely rigorous, I have to really prove that if I will cut the cone and open it up, roll it up on the surface, first of all, that it's rollable on the surface, on the flat surface. I mean, there are certain surfaces which are not rollable on the flat surface, for instance, a sphere. No matter how you cut the sphere, you will not be able to completely flatten it, right? But the cone you can, cylinder also you can. Again, you cut it and roll it out. But that's because the cone and the circle are constructed from a conical or cylindrical surface, each of them contains straight lines. So, that's important. Well, not necessarily every geometric figure which contains only straight lines can be flattened. That's a different story. I mean, there are certain geometrical figures, geometrical objects which contain only straight lines, but they are not flattable. Now, this one is, but that needs to be proven a little bit more rigorously, which I'm not going to do because it's a really complicated stuff. And the result of this would be exactly a sector as I was just picturing it. So, that's kind of the thing which is intuitively obvious, and that's where I would like to stop. Because, you see, in many cases, mathematics has been developed first based on intuition and only then the rigorous proofs were actually offered. So, it's very important and that's why I'm very much encouraging you to use your intuition in cases like this, for instance. I mean, sometimes you might be wrong that that's true. But the more you go through all these processes and the more baggage you have as the foundation of your intuition, the better your intuition will be. So, in this particular case again, I rely in the beginning on intuition and the calculations are simple, obviously. Now, let's talk about the volume. So, the volume of this cone is actually very quickly, which I can explain what it is. And then I will again talk about philosophy of this. Okay, so what we can do is let's inscribe, let's say, a regular polygon into this base circle. It would be something like this. Let's consider this to be a regular polygon inscribed. And I will connect every vertex of this polygon. Now, I will get a pyramid, right? Now, pyramid would be inscribed into a cone, obviously, you understand. Now, obviously, as I increase the number of vertices on this circle in such a way that even the maximum edge is going down to zero, for instance, I can always double the number of edges by dropping a perpendicular from the center to each edge of the polygon and replace one edge with two in between. So, I will double the number of edges. Obviously, this process is doubling the number of vertices, diminishing down to zero the length of each edge, which means that our polygon will be closer and closer to a circumference of a circle. So, first it will be like this, then it will be like this. So, the closer it becomes, the closer the volume of the pyramid would be to the volume of the cone. I mean, that's kind of an assumption, right? It's not a logical assumption. And in the limit, we can say that the limit of this volume of the pyramid would be the volume of the cone. I mean, it's kind of a natural, obvious, intuitive, obvious thing. Well, now, we know that the volume of the pyramid is equal to one-third area of the base times h, right? Now, all these pyramids have the same altitude as our cone. So, this is a constant. Now, the area of the base is basically an area of this polygon. And as we increase the number of vertices and the edge is going down to zero of each edge, this area of the base would go to area of the circle, right? Which is pi r squared. That actually leads us to an assumption that the whole volume is equal to one-third pi r squared h. That's the volume of the cone. Now, again, what's not very rigid in this particular case? Well, this process, which I was just indicating, it's not such a simple thing. And obviously, we have to prove that as we increase the number of vertices with every edge diminishing down to zero, then, well, then obviously, the variable which we are generating this way, this one, the volume of this pyramid, it's changing and it has a certain limit. So, we have to prove existence of this limit. Secondly, what we have to prove is that no matter how we arrange this process, maybe we can start with a square and then double the number of sides, or we can start with a hexagon and then double the number of sides, or we start with a triangle, for instance, and we triple the number of sides, something like this. You start with a triangle and then you triple the number of sides instead of this. And here, instead of this, et cetera. I mean, there are many different ways you can increase the number of vertices and decrease the every edge of this polygon. So, it looks like it tends to the circle, right? Now, will every of these ways lead to the same limit? So, the limit should be unique. So, there are certain theorems which must be really proven rigorously if you want to use. And again, all I'm saying right now is that intuitively it's obvious that the area of this polygon would tend to the area of a circle. However, no matter how intuitively obvious it is, the rigorousness is needed. Like, for instance, in this particular case, let's just think about what happens if you are increasing the number of vertices. Well, you are making each line closer and closer to the surface, to the circle it's inscribed into, right? So, the difference becomes smaller. But the number of these differences is corresponding to the number of sides. So, every little difference becomes smaller, but the number of differences is increasing. The question is whether the whole difference between the area of the polygon and the area of the circle really goes down to zero. It needs to be proven. So, it's true, and again, intuitively it is obvious, but these things need to be rigorously proven, which is outside the scope of this course. I'm really very much in leaving this particular theory as it is right now, without going into any more rigorous calculations. But anyway, these intuitively obvious logical assumptions lead us to this very obvious formula for the bottom of the code. Well, that's it for the code. I do suggest you to read the notes for this particular lecture. They are on Unizor.com website. And, well, that's it for today. Thank you very much and good luck.