 Hi, I'm Zor. Welcome to Unizor education. Today's lecture will be about truncated pyramids. It's a continuation of the pyramids topic. Now, obviously this is part of the course of advanced mathematics for high school students. It's presented on Unizor.com. That's where I suggested to which this lecture from because there are notes and in this particular lecture actually the notes are quite extensive. So I do suggest you to look at the notes before or after listening to this lecture. Alright, so this is about truncated pyramid and first we have to explain what is truncated pyramid. Well, that's actually quite simple. You have let's say a pyramid like this quadrilateral pyramid with some kind of a quadrilateral at the base. Well truncated means we just cut off the top of it, basically. That's it. And whatever is left is truncated pyramid. Now, let me be a little bit more specific. So let's say we have pyramid like this. Now next what I will do is I pick any point, let's say this one a prime on SA H and draw a plane parallel to the base plane. Let's say this is the base plane and now I'm cutting with a plane parallel to beta. Let's call it gamma and as a result my pyramid will be cut like this something like this. That's what plane gamma will cut off the pyramid. Now this is supposed to be invisible. Right? Okay. And intersection of this plane with SB I will call B prime with C, C prime and with D, D prime. So A, B, C, D, A prime, B prime, C prime, D prime. This figure which is left after right cut off the top using this plane parallel to the base. This is called a truncated pyramid. That's the definition. Now very elementary properties. First of all, since this plane is parallel to this one and you can consider S, A, B plane as truncating these two and you know the theory if you have two parallel planes and you have another one which transcends both of them then the resulting intersections of this plane with the parallel planes will be parallel lines. So A, B and A prime, B prime are parallel to each other because this plane is parallel to this plane and this is the plane which intersects them both. Similarly, B, C parallel to B prime, C prime, etc. So all corresponding edges will be parallel to each other. Which in particular means that A, B, B prime, A prime is a trapezoid, obviously, as well as all other sides. So sides, side faces of truncated pyramid are trapezoids, right? That's that simple. Now, let's consider triangle S, A prime, B prime and S, A, B. Now, since A, B and A, B prime are parallel to each other, these two triangles are similar with the scaling center at S, at apex and the factor basically being S, A, divided by S, A prime. Now, that's exactly the same factor as between S, B and S, B prime and now if you consider triangle S, B, C and S, B prime, C prime, you see that the S, C and S, S, C and S, C prime also have the same relationship, also are of the factor F. So this factor F is shared among all of these guys because the factor of this is equal to factor of this, factor of this is equal to factor of these and factor of these is equal to factor of S, G and S, D prime. So they're all the same and same exactly ratio for A, B, relative to A, B prime. So this is a very interesting situation. Everything in the small pyramid and the large pyramid is proportional with the same scaling factor F and center S. In particular, it means that not only every linear is proportional, also the errors of corresponding pieces of these pyramids are proportional, but the proportionality is F square. Remember this? So the area of let's say A, B, C, D divided by area of A prime, D prime, C prime, D prime is equal to F square. So remember when we were talking about similarity in the three-dimensional world, linear elements are proportional to some factor and aerial quantities, like area of a triangle or area of quadrilateral, they are proportional to F square. We did discuss this. So that's very important because we will use it obviously. Now what else is important here? Okay, so let's draw an altitude from S down to H and obviously intersects the top quadrilateral as well at point H prime. Now my statement is that H S H ratio to S H prime is also the same F. Now, how can I prove that? Well, it's really very easy. It's basically, you can consider S A H plane which intersects two parallel planes, beta and gamma and that's why A H is parallel to H A prime H prime and therefore they are proportional and proportionality is exactly the same as S A to S H to S A prime. So again triangle S A H and S A prime H prime are similar and you have the same proportionality between these two lines as before. So it extends to to these. Now S H, I will call lower case H and S H prime lower case H prime. These are altitudes of the big and the small pyramid. All right. And these two guys area of the basis, I will call lower case S and upper case S. I mean lower case S and lower case S prime. Okay, so right now I would like to concentrate on these two. So we have big and small pyramid. We have their areas, areas of their basis related as F square and their altitudes are related as as plane F. Now, I think I have mentioned everything to start talking about the volume of the truncated pyramid. Now, what's very easy right now is to say that the volume of the truncated pyramid is basically the difference between the volume of the big pyramid and the small pyramid which I'm cutting out. So basically V is equal to one-third S H minus one-third S prime H prime. Where S is area of the ABCD, H is an altitude of the pyramid, S prime is area of A prime, B prime, C prime, D prime and H prime is the altitude of that small pyramid. That's fine, but it's actually it's not good enough. And let me explain you why. Now, you see if we are talking about truncated pyramid and you would like to express its volume in some terms, I better use the terms which belong to truncated pyramid, not to some composition of this truncated pyramid and something on the top which I don't really have, etc. So I would like to express the volume of the truncated pyramid in terms of elements of the truncated pyramid, not the elements of the pyramid which does not exist actually. So, what does exist? Well, what does exist is the the altitude of the truncated pyramid itself, H H prime. Let's call it A. So, what I would like to express the volume, not in terms of S, which is a characteristic of a truncated pyramid, S prime, which is also a characteristic of a truncated pyramid and not in terms of H and H prime, but in terms of A, which is basically the height of the pyramid itself, which happens to be, by the way, H minus H prime, right? So, the height of the truncated pyramid is the difference between the height of the big pyramid and the small pyramid. So, I don't like this. I would like to express instead of B in terms of S, S prime and A, not H and H prime. Question is, is it possible? Well, let's just think about it. I don't really need this because right now we are in the algebra world. Look at this. This one, this one. Okay. So, we have proportionality between areas, proportionality between this. Now, let's consider these three guys. What's known? What's unknown? I would like to express the volume in terms of S, S prime and A. So, these are kind of known variables. F is not known, H and H prime is not known. So, we have three unknowns and three equations. So, what if I will solve these equations and I will get the values of H and H prime in terms of S, S prime and A and substituted here? Well, that would be my final result and final formula where my volume is expressed in terms of S, S, S prime and A. No H and no H prime. So, that's all I want to do. I would like to exclude H and H prime from this formula using these three equations with three variables which I can solve. Solve for F and H and H prime, right? Well, okay. So, for F actually, it's very simple. It's square root of S divided by S prime. So, that's it. And I don't really need this equation anymore. Also, what I will do, I will express A plus H1. I will express S as A plus H prime. So, what I can do right now is I can actually substitute. Now, from here, I can actually take, if I will put H prime here instead of H, that would be F is equal to A plus H prime over H prime, which is equal to H prime plus 1, right? From which I can derive H prime is equal to F minus 1, A divided by F minus 1, but I know F. So, that's this. And now I can do this. So, you see, I have H and H prime expressed in terms of A and S and S prime. So, that's all I need, actually. I have to substitute into this formula. Well, and go to simplify it somehow if I can, right? So, let's do it. So, what do I need? I really need this line and this line. I don't need F anymore because F does not participate in my volume formula. So, the only thing is, let me just simplify it a little bit, this one. H is equal to, if I will use common denominator, it would be A times square root minus A and plus A. So, it would be A square root of S over S prime divided by square root of S over S prime minus 1. So, these are two values for H and H prime, which I'm going to substitute into this formula. And what do I have? V equals 1 third S H, which is A square root of S over S prime divided by square root of S over S prime minus 1. Minus 1 third of S prime H prime, which is A over the same denominator. So, that's all I have to simplify. So, from now on it's just technicality. Okay, so what do I do? Well, let me just make very quick thing. So, I will multiply by square root of S prime, both denominator and denominator. So, what I will have is, now A over 3 would be outside. A and A and 1 third and 1 third is all outside the parenthesis. Now, inside I will have, so if I will multiply by square root of S prime, I will have, on the top I will have S square root of S. On the bottom I will have square root of S minus square root of S prime, right? If I multiply by square root of S prime, this is this and this will be this. Minus and denominator actually is the same, so I can use the same here. S prime, I multiply by square root of S prime and denominator will be the same as this one. So, that's basically my result, right? Well, I can obviously simplify it even more because very simple thing. Well, probably the easiest way to explain is this one. If I will put like this, what do I have in this case? S square root of S is basically X cube, right? And this one is Y cube and this one is X minus Y. Now, this should be familiar because X cube minus Y cube is actually a multiple of X minus Y times X square plus X Y plus Y square. If you don't believe me or if you don't remember this formula, just multiply this by this and you will get this. X cube plus X square Y plus X Y square minus X Y square and minus Y cube, so that will be what it is. Everything else will be reduced, cancelled out. So, what is this? X cube minus Y cube divided by X minus Y is this one. So, my result is A divided by 3 and in parenthesis I will have X square which is S plus X Y which is square root of SS prime and Y square S prime. And this is the final formula for the volume of the truncated pyramid. It's one-third of the altitude and then you should multiply it by this expression where S and S prime are areas of two bases of the pyramid. Well, that's it. I would suggest you to read the notes for this lecture because they basically explain the same thing and all the formulas and transformations are there. Well, basically, that's it. Thank you very much and good luck.