 Hi, I'm Zor. Welcome to InDesert Education. Previous lecture was dedicated to spherical caps, and I was discussing the volume. Now, one of the very rare occasions when I can say that certain mathematical properties can have a direct implication onto reality is that the surface, the top surface of the spherical cap, which I call a dome, is something which people do see in architecture, in cathedrals, for instance. And obvious question is what's the error? I mean, for instance, you have to cover it with something, and you have to know how much material you have to have. So, to satisfy these practical implementation, I would like to discuss today a surface of the dome of the spherical cap. However, before doing that, I would like to introduce spherical sector first and its volume, and then I will show how easily I can get the surface of the dome from volume of a spherical sector. In some way, it will be very much like getting a surface of a sphere from its volume. If you remember, I was using certain technique which I will use exactly the same thing today. Okay, so what is a spherical volume, a spherical sector? Sorry. Well, if you have a sphere, my artistic abilities will definitely be challenged here. Now, if this is a cap, a spherical cap, which is the result of cutting the plane, then this is a spherical sector. It contains the cap from the top and the cone on the bottom. So, when the cone actually meets the cap, you have this circle which is an intersection between the plane and the sphere. Now, this is a very small addition to our spherical cap. So, all I have to add, if I want to calculate the volume, I have to add the cone. Now, I know that this cone is a right circular cone. The radius of the cone, let's call it L, L squared is equal to Pythagorean theorem. This is R, so it's R squared minus, and this piece is the radius minus the height of the cap, right? So, I know that. Now, the altitude of this cone is basically R minus H, right? So, the volume of the cone is equal to 1 third pi L squared. That gives me the area of the base of the cone times its altitude, which is this. And if I will substitute L squared here, I will get the volume of the cone is equal to 1 third pi. If I will open the parentheses, it will be 2RH minus H squared, right? Because R squared and R squared will cancel out each other. So, it's minus 2RH, so it's plus because there is a minus and then plus H squared, which is minus because... So, instead of L squared, I will substitute this 2RH minus H squared times R minus H, and I can open parentheses and see what happens. By the way, H can go outside. The volume of the cone is equal to 1 third pi H... Excuse me. Okay, I will have 2R minus H times R minus H, okay? And that would be equal to 1 third pi H 2R squared minus HR minus 2HR plus H squared, right? Which is 1 third pi H 2R squared minus 3HR plus H squared. Okay, that's my formula for the volume of the cone. Now, if I want the volume of a sector, I have to add the cap, right? Now, the volume of the cap from the previous lecture was, I have it written somewhere, 1 third pi H squared 3R minus H. Yeah, that was a previous lecture. Now, the volume of sector is equal to some of these, right? So, 1 third pi H 2R squared minus 3HR plus H squared. Now, I have only 1H outside of the parentheses, which should be plus 3HR and minus H squared. Which is what? 3HR, 3HR, H squared, H squared. And the final volume is very simple formula. 2 third pi R squared H. See, after all these very unpleasant kind of formulas, the result of the sector is really very nice and short formula, which shows that the sector is, well, I would say, well, you know, there is a, I think Einstein was one of the people who had beauty as a criteria for truth of the hypothesis or something like this or theory. Well, so the nice formula actually signifies that this is a very natural kind of geometric figure. All right. Now, what we have to do now is I promised you to calculate the area of the dome, this area. Now, if you remember the way how I did it with volume of the sphere and the surface area of the sphere, I have suggested to cover the dome with polygons, very small polygons, which are approximating, they are like inscribed into this dome. So let's say I have chosen a certain number of points here and connected those points with segments. Like every three points, I connect with a really small triangle, right? So every three neighboring points are connected with a triangle and then every point I connect to a center. So a triangle and a center give me the triangular pyramid. Now, if I will increase the number of points, my triangles will be smaller and smaller and they will approximate the area of the dome tighter and tighter. Now, if I want to calculate the volume of all the triangular pyramids, I have to do what? I have to summarize one third area of the pyramid's base times its height. Now, the height will be closer and closer to R, right? So that's like a constant and some of all these areas of the pyramids will go to the area of the dome, right? So the volume of the sector is equal to one third area of the dome times radius. So that's the formula which I have not proved, but from the intuitive considerations which I have just explained, this is the right formula, okay? Now, from this formula and knowing what is the volume of the sector, which is two third by R squared H, I can derive that the area of the dome of the sector or a spherical cap, whatever you want to say, is equal to F to divide by R, one third here and here disappears. So it's two pi R H. So, if you are a Michelangelo or whoever else was building cathedrals some time ago, that's your formula for calculating the area of a dome if you know the radius and the height of this cap on the top. Well, I do suggest you to repeat all these calculations yourself independently. I mean, you can look at the notes to lecture. And it's a good exercise. And again, considering you have such concise formulas, it just signifies that everything is right according to this theory of beauty. So thanks very much for your attention and good luck.