 In this video, we are going to consider some thoughts and methods about how we might approximate the volume of a t-ondang fire lantern given some measurements that we take on the sides of the lantern. So you may have seen in a previous video the actual measurements, the taking of the measurements, in which case what we've done is we've taken the fire lantern. Here's an example. And we've extended it and recognized that the fire lantern is created by four different sides glued together. And each of these sides creates this kind of petal shape. And what we've done is we've measured from the top of the petal shape down to the bottom of the petal shape. And we've done some slices across to give us some estimation of the shape of the geometry of the overall lantern. So here, I've made an effort to sort of sketch on a smaller scale the shape that we have, along with the measurements that we've been taking. And then I've also done my best to sort of show what it looks like when we take the lantern and we let it inflate. When we do so, instead of being this vertical shape, we're going to sort of see that the entire lantern expands and fills. The shape compresses a little bit and we create three-dimensional shapes out of the two-dimensional shapes here by having one of those shapes on each of the four sides. So we've taken a series of measurements. Those measurements are along this vertical axis along one of the sides. And we're going to label those as a series of heights. And in this case, we're going to call this H0, H1, H2, H3, H4, and H5. Or in other words, that series is H sub i, heights that start at 0 and count up. And for each of those heights, we have a corresponding width that goes all the way across the lantern. And we'll label those W0, W1, W2, W3, W4, and W5. Or our Wi, where i is just a generic index indicating one of those points. Notice you can see where each of those sections are over here in the three-dimensional view. There's one of my widths going across. I guess this would probably be width number 4. And notice that's located at height number 4, H4. However, an important thing to notice is that the height up the middle of the lantern is different than the height that we've measured here. Notice the curvature that occurs here going from H0 to H1 is actually this distance here, which is at an angle once the entire t-undang is inflated. This would be from H1 to H2. I guess I've mislabeled this as H4. This would be H3, H2 to H3, H3 to H4, and then H4 to H5. So that's a very important concept that we want to keep here is that what we've called height for our original measurement isn't exactly a height. It's a curvature around the outside once we've inflated the lantern. And then we have these associated widths that go along each piece. So we're going to make an assumption here. An approximation of each piece based on this idea of us slicing across. And this approximation will not be perfect. There are better ones. More accuracy equals more work. Whenever we're doing computational approximation, we could do more work on different things. We could think more theoretically and have more detail in how we actually describe this. Or we could slice this up into smaller pieces, many smaller pieces, and then our approximations and our geometry would be less and less important because we would be closer and closer to a reality. But what we're going to do here is we're going to assume that each of these pieces, take a look at this top piece up here, for example, that each of these pieces can be represented by a particular shape known as a frustrum. Now, you were probably all familiar with the frustrum from when you were kids. If you ever had an ice cream cone, one of the long, thin ice cream cones that come down to a cone at the bottom. And at some point when you were eating your ice cream cone, it started melting into the cone and you probably bit the bottom right off of the cone so you could slip the ice cream out of the bottom. A lot of us have done that when we were kids. Well, that creates a shape for you, which is a cone without the tip. That's an example of a frustrum. That same thing also applies to a shape that might be like a pyramid. In this case, I'm going to draw the pyramid right side up. If you have a pyramidal shape and you chop off the top, that's going to create a frustrum of a pyramid. And notice you're going to get some sort of shape at the top. So we're going to assume that each of these pieces that we see here can be approximated with this frustrum, a frustrum of a pyramid. Let me go ahead and sketch that for you here. So we'll take an example here. I'm going to go ahead and there's the top plane. And down here we'll have a bottom plane and it will be connected on the sides. All right, and again, we're thinking about this in three dimensions. So maybe I could have made that a little bit flatter. Okay, and along each of these we're going to consider, here is a width and we'll call this width at some space I, width I, and then we're going to have a width right above it of width I plus one. So we're going to, when we're looking at any frustrum, it's going to be between two of our measurements, one that's below and one that's up above. Actually, let me actually do it this way. I'm going to do width I and width I minus one. So we're going to consider the space I and the space right below it instead and that'll be a little clearer why in just a second. Okay, so for each of those, we are also going to have a location height I minus one and a location height I. But again, notice that location is kind of up around the sides, up around the sides here. We are going to go ahead and call this piece of the frustrum, volume, we're going to try to figure out the volume of this piece, volume I. And the reason why I did I and I minus one is because down here you see where it says H zero, my height zero, well we're going to call the area of the volume of this one volume one, volume two, volume three, volume four, volume five. So we're starting our volume count at number one. So notice our volume for number one uses the information from one and zero. Volume from number two uses from two and one, volume from three, three and two and so on. Okay, so now we have this geometry of a frustrum that we're interested in. But when I do my research on a frustrum, generally what they consider, and we'll look at the formula for a frustrum in a little bit, if I can sort of consider what's happening in the back in my third dimension, generally what we're interested in the frustrum is the actual height of the frustrum, but that height perpendicular to those two planes. So a height up the middle of that frustrum. We're interested in some height value here that runs right up the middle of the frustrum. I'm going to identify that with a capital H to distinguish it from our heights that we measured around the side. This is sort of an important conceptual piece here that we're measuring this line up the middle of the thing when it's expanded, which is different than the heights we measured up around the outside pedal, the outsides of the Tian Deng. All right, well to do that, what is that height? What's it going to look like? What is the value of that particular height? So I'm going to go ahead and look at the trapezoid. There's a trapezoid over here on each side, or I'm going to think about this as being, okay, what if I slice this frustrum in half, and it's going to create for me a trapezoid shape that runs kind of across, but has that height up the middle. So I'm going to recreate that trapezoid here. There's the trapezoid shape that we have, and that's a slice right down the middle of my frustrum. And we'll notice that across the top of my frustrum, there's my width, width I, and along the bottom of my frustrum is width I minus one. So that information is there and useful for us. What other information do we have? I'm trying to figure out this height here, h i. Well, if I look at the sides of this trapezoid, those sides are running along the face of this thing. I actually was probably a little bit strange to identify this as h i and h i minus one, because those are kind of the corners of each piece. Well, I'm going to assume that this thing kind of puffs out all the way around. And again, it's an approximation, but I'm going to call this distance here this change in height is my difference between h i and h i minus one. That's what I'm going to say that is there. Now notice that's a little bit different. It's kind of this slant here, as opposed to the slant on the diagonal. We have this and then we also have that, but I'm assuming we're kind of puffing around almost as if we're in a cone. And again, it's an approximation. So if I have this height here, delta h, and I'm trying to find this vertical height, how can I do that? Well, if I look, I now have a right triangle. There's my right triangle. And I notice that this little piece here, if we assume some symmetry is half, one half of the difference between my widths. So now I have enough geometry to make an estimate for the height of my frustrum. So now that we've created some geometric relationships here, let's go ahead and write down what these are. Using the Pythagorean theorem and doing a little bit of algebra, we can determine that my height of any of my frustrum is going to be equal to some relationship between the different heights and the different widths that I've already measured. In that case, my formula looks something like this. And this will look familiar from a squared plus b squared equals c squared or from your distance formula, but notice we'll be using a subtraction in here because we have the hypotenuse and we're looking for one of the legs of my right triangle. So it's gonna look something like this. We're going to have our height minus the height before it squared. And then we are similarly going to subtract off the weight minus the weight before it squared. However, because this was only half of that, we're going to have a divided by four here. Why do we have a divided by four here? Well, because we have a half, but then we're squaring that entire thing. So then we use that half twice and there's the factor of four. So now we have the formula for the height of our frustrum. Well, that means we will be able to calculate this height from the information we have. Now, the next part is what is the volume of a frustrum? So to find the volume of a frustrum, we're gonna go back to ancient Greek history and look for something known as the Heronian mean. Heron of Alexandria created something known as the Heronian mean. And this is going to be a relationship between the top of our frustrum and the bottom of our frustrum. And we can use this Heronian mean to figure out the volume of the frustrum. Here's what that formula looks like. First of all, in any volume, we're gonna have some sort of height times maybe your other two dimensions. So indeed, here we have the height value. And then we're going to have something that's an area. Well, in this case, it's gonna be an average area of these two parts of the top and the bottom. And it actually looks something like this. It's gonna be the area of the upper part. Here's area I on top. And we're also going to add the area of the lower part. There's the lower part. But here's where the Heronian mean comes in. There's also a third piece that takes both into account. And it's a very interesting-looking piece. It looks something like this. There's a square root. The square root of what? Well, the square root of the product of the two areas. So there's my Heronian mean. There's an area. There's the square root of two areas multiplied by each other, which is also an area. And then there is an area. And that takes into account both the top and the bottom. There's one more piece here. Notice we added three things to find this mean. So we're gonna go ahead and make it a true mean by dividing by how many things we added together. So again, the top area, the bottom area, and then this Heronian part, which is a way of averaging the two areas or combining the two areas together in a way that is consistent. It's kind of like basically taking two rectangles or rectangle in that particular case. So that's great, except we don't have areas here. Or do we? If we look at these, each of these areas is simply a width times a width or the width squared. So now we have our heights in terms of our initial measurements, height and width. That's our height at the middle. And we also have these areas that we can determine from all those widths. So therefore, we have all the information we need. If we've measured those things, we have all the information we need to calculate the volume using the Heronian mean. If we want to, we can simplify the volume formula and write it simply in terms of those widths. One third will keep this height value since it's fairly complex. But I can replace the a's with the w i's with the widths. There's w i squared over here. We have w i minus one squared. And then if I take the two of them and multiply them together, I get w i w i minus one, the product of the two of them. Notice they're each squared, but then we square root so the squares go away. And again, we end up with basically a square for the top, a square for the bottom, and then effectively a rectangle. That is the average of the two of them. And that's kind of how the Heronian mean should be thought about. So we now have the means that if we've measured along the sides, this series of heights and widths along the sides of the Tian Deng, we now have the means of finding the volume of each of those pieces. Our last piece is then to find the total volume is to add up all of those other volumes. There's my sum meaning to add up as I goes from one to however many we have of all the other volumes. That formula we will use in the next video to go ahead and show an example of calculating the volume, of approximating the volume using calculation of a Tian Deng lantern.