Common Core Geometry Unit 10 Lesson 11 The Volume of a Truncated Cone

So what's a truncated cone? A truncated cone is what we get when we take a cone, we slice it horizontally, and then we throw that portion away. So what do we have left? Well, we have what most of us would think of as a cup. All right? So a truncated cone is an extremely, extremely common geometric shape. You know, you'll see them everywhere. The question is, how could we figure out what the volume? Of this cup is. So let's start to explore that problem. All right? Exercise number one, right? Here we've got that picture of a truncated cone. What strategy might make sense to find the volume of the cup, the truncated cone. What strategy might make sense to find that volume. Pause the video now, think about this, discuss it with your classmates with your teacher, et cetera, and then let's come back to it. All right. Well, hopefully after the problems that we've done with shaded areas and removing volumes from other volumes and things like that, hopefully the strategy should be pretty clear. If we can calculate the volume of the large cone, the big cone. 

And we can calculate the volume of the smaller cone on the bottom section, then we can simply subtract the volume of the smaller cone from the volume of the overall cone, and what will be left with is the cup. Because isn't that really what subtraction is. If we could literally have this solid, if this was a solid piece of something, and we wanted to know the volume of what we had up here, we'd simply take the overall volume, and we would subtract or cut off that lower portion. So we're going to find. The volume. Of the two cones. And subtract. The smaller. From and had to happen at least one more time. From the larger. Maybe we'll even just leave the red there. All right, that's it. It's very simple. So the hole is the sum of its parts. The entire hole, right? Is the sum of the upper part, which is actually what we care about plus that lower part. 

Now, exercise two. What information would you need to carry out the strategy from one? So what would I have to tell you in order for you to do this calculation? Again, pause the video and talk this over. All right. Well, to really properly answer this question, we have to be comfortable with the volume formula for a cone. So let's remember what that is, right? If we have any cone whatsoever, that's, well, let's say a circular cone. All right. Its volume is one third, pi times R squared times H and just to make sure that we remember all of this, right? The pi R squared is simply the area of that circle. All right, the height, well, is that distance. And remember, the one third is there because we're dealing with a cone instead of a cylinder. All right, if we had a cylinder, then the one third wouldn't be there at all. And it would just be pi R squared H so we need to know the radius and the height of any cone in order to calculate its volume, which means in order to carry out this particular strategy to find the volume of the truncated cone, well, we need to know the radius of the large cone. Right? So we need to know radius of the large cone. 

We need to know the radius of the small cone. We need to know the height. Of the large cone. And we need to know the height. Of the small cone. If we have those four things, we can calculate the volume of both cones. They'll be messy, right? Because we've got fractions, one third, we've got pi and a rational number. We've got squaring. We've got a lot of different things going on. But we also have our calculators. So at the end of the day, we just need these four values. All right? So, let's keep going and actually do one of these problems. All right, here we go. Say we have the following cup, which is 16 centimeters tall. I would like to pause just for a minute. When you look at all the dimensions sitting here and I've given you with a cross sectional view of this cone, right? I'm telling you that the cup itself is 16 centimeters high. And of course I can figure that out because I've got this height, which is 28 centimeters, this height, which is 12 centimeters, so that would have to be 16 centimeters. 

Anyhow, we have a cup, which is 16 centimeters tall. It has a large radius of 7 centimeters, and it's got a smaller radius of three centimeters. Letter a, asks us to find the volume of the cup to the nearest cubic centimeter, show the work that leads to your answer. All right. And again, the strategy that we're going to use is the one that we laid out in the first exercise. We're going to figure out the volume of that large cone. Then we're going to figure out the volume of that smaller cone, and then we're simply going to subtract the smaller volume from the larger volume, leaving us the volume of the cup. So let's just start to write those things out, right? The volume of the larger cone will be one third times pi times R squared times H, we know absolutely everything. One third pi times what's its radius at 7. Squared times its height, which is 28. All right. So we've got to do that on our calculator. That's going to be a messy calculation, all right? We'll get the answer to that in a moment. The volume of the smaller cone will be one third pi R squared H and I would encourage you to keep writing that formula down. 

Every time you use it, simply because it's going to make it easier to substitute in the numbers. Right? So that's also going to be some messy number. And again, putting these into your calculator is what it is. So what I'd like you to do now is pause the video for a minute and work this out on your calculator. While you're doing that, I'm going to fire my calculator up, which I don't think I've done yet this morning. So go ahead and work those out. All right, hopefully you're all done. We've got the calculator up and running. I'm just going to work through one of these to illustrate it on the calculator that I'm going to write the other one down so you can check your answers, and then we'll do the subtraction. It's going to be as simple as that. So why don't we do the one third pi times 7 squared times 28? Again, we just have to be careful. One third, we don't even have to put the time symbol there, but I'm going to put it there anyway. 

One third times pi times 7 squared times 28. All right? So we've got the one third pi, the 7 was our radius squared times 28. And we're going to get an awfully large number. One 41,436 point let's call it 8 cubic centimeters. All right? We could store that in something if we wanted to. We could store it in letter a or letter B or something to help us out later. I'm going to write it down in just a moment. 1436.8. And cubic centimeters. So that's that volume of the large cone. Let me just verify that really quick. Absolutely. Great. And the volume of the smaller cone. I worked out previously 113.1. By the way, that might seem much, much smaller, and it is. All right? And it's much smaller because we've got a three squared, which gives us a 9 versus a 7 squared, which gives us a 49. And of course, those 12 and 28, those also compare. So to get the volume of the cup, simple enough, all we're going to do, let's go back into this now. We're simply going to take those two volumes. Let me call it volume C for cup. 

That's the volume of the larger cone minus the volume of the smaller cone. Simply 1436.8 -113.1 hello red. And we end up getting 1324. Cubic centimeters. All right, that's it. So it's not that hard, right? I mean, it's annoying, right? Because you've got all this kind of messy calculation to do, but that's often just the case with volume problems in general. I mean, think about if you were with us for the last lesson, when we were doing the calories of that ice cream cone. I mean, that thing was messy. We had three different volumes. We had to do a bunch of different conversions. Don't get scared by messy. It's just being careful with your work. All right, now this is very important. Letter B asks us to find the ratio of the two cone heights in simplest form. I want to find the ratio of the two cone heights in simplest form. All right? And why don't we go ahead and do that together? I know it's a little bit small on the screen. But one of the heights is 28, and one of the heights is 12. 

Now, you could certainly do that in the opposite you could do 12 to 28, or you could do 28 to 12. It doesn't particularly matter. And what I'd like to do is reduce that now. So pause the video really quickly and just using your head reduce 28 12 down to its simplest terms. All right. Well, let's do it. You know, obviously when you're reducing fractions, what you do is you divide both the numerator and the denominator by the largest factor of each that's common, and we end up getting a ratio of 7 thirds. Now, the question then, why is this ratio? The same as the ratio of the two radii. So notice what we're seeing here is we're seeing that the radius of the large divided by the radius of the small is also 7 thirds. So why would those two ratios be the same? Pause the video now and discuss this. All right, well, any time you're looking at equivalent ratios, that typically has something to do with similarity. And this doesn't just have something to do with similarity. This has everything to do with similarity. 

You see, remember, a cone is created. At least a right circular cone is created. By taking a right triangle and revolving it around sort of that center axis. All right? This pair of right triangles. And yes, it's a pair of right triangles, right? One of the right triangles is this large one. Right? That larger right triangle. And one of them, just to go maybe in a little bit of a different color. Is this smaller one? And those two right triangles must be similar to each other. Think about it. We could prove that they're similar to each other using the angle angle similarity theorem. They both have a right angle there and there. And then they both share this angle down here. So these two triangles are similar to one another. And because they're similar to one another, right? Ratios of corresponding sides are equal. So this side is to this side as this side is to this side. Now, that's going to become eventually quite important, but let me go in and just fill this in really quickly. Okay. Ratios are equal. And we'll just say due to similar right triangles. The return of the ghost pen. All right. Now, one last little piece before we move on to a more challenging version of this problem. 

And this is extremely important. Because some of you, I hope, have been asking this exact question in your head. Remember, the bigger piece today is that we want to be able to calculate the volume of that cup, right? Right here. Of the measurements given in this problem, here they are. Which one is unrealistic that we would know. So if I just I've just got this cup. And I want to figure out what its volume is. What seems unrealistic about having this information. What wouldn't we typically know? Well, we typically wouldn't know the height of the smaller cone. We also by extension wouldn't not have the height of this, but the more problematic pieces we would be able to know that height, but we would never know what this height is, because it's not there, right? Literally, on this cup, let me go over to bread so that you can see it a little bit better. On this cup, that portion. Is down here, and it's not actually there. I mean, mathematically, it's there, but it's not physically there. There's no way to really take out a ruler and go, hey, that thing is 12 centimeters. There's just no way. So we wouldn't know. We wouldn't know the height. Of the smaller cone. There's no way to know it. Well, okay, I shouldn't say there's no way to know it. 

But we certainly wouldn't know it up front, right? What would we know? Well, we'd know the height of the cup. We could certainly measure than the radius or the diameter of the top of the cup when we could measure the radius or the diameter of the bottom of the cup. So every other piece of information on this diagram, we would know. We just wouldn't know this. So next, we're going to figure out how to deal with that. All right, here we go, exercise number four. You ready? In the following cut problem, a cup with a height of 18 centimeters has a large radius of 5.4 centimeters, and a small radius of three centimeters. Set up and solve an equation that could be used to find the value of X, the height of the smaller cone. So again, this is the piece that we would never know. Now granted, we also don't know the height of sort of the larger cone, but once we know the height of the smaller cone, we'll simply add it to the height of the cup, and then we'll be good. So how do we set up an equation to do that? Well, again, keep in mind that this larger right triangle is similar to this smaller right triangle. So pause the video now and see if you can set up and solve an equation to figure out what the height of that smaller cone is. 

All right. Well, if we're dealing with ratios of similar right triangles or similar triangles in general, we want to make sure that we have the side lengths of those triangles. So if we know that's 18 and we know this is X, then this overall height must be X plus 18. A little bit tricky there, but we got it, right? X plus 18. So now what do I have? I've really got these two similar right triangles. One of them looks like this 5.4 and X plus 18. And the other one looks like this. Three and X so I can now set up a variety of different ratios to solve for X I think for me what I'm going to do is I'm going to say 5.4 divided by three, that ratio, 5.4 is to three. As X plus 18 is to X, all right? That's it. Now I have to solve that equation. All right? Can I move this all at once? Oh, I can. Oh, now I can't, now they're ungrouped. Oh well, I think that gave me enough room. Hopefully you can still see it. But yeah, let's now play around with this. I'm just going to cross multiply. All right, we should feel comfortable with how to do that. All right, that's going to give me 5.4 times X is equal to three times X plus three times 18. And three times 18 last I checked is 54. Yep. All right. 

Now I have to solve this. Don't be scared by the decimal. Whatever, I'll just subtract a three X from both sides. And of course, if I have 5.4 something and I subtract three of the same thing from it, I end up with 2.4 X is equal to 54. And then I can just divide both sides by 2.4, these problems are oftentimes not very nice. And we could do that on our calculator, but of course I've got it all worked out. We got 22.5. All right, 22.5 centimeters, okay? Keep your eye on the ball. We just solved, right? We just solved for the height of the smaller cone. So we have everything we need now. The one thing we didn't have was the height of that smaller cone. Now, we also need the height of the larger cone. Okay. But we'll fill that in as we now come down to the next part of the problem. All right, we'll leave that 22.5 right up there at the top. Letter B one liter is equivalent to a thousand cubic centimeters. Can this cup hold more than one liter? Justify your answer. What you should have everything now to figure out the volume of this cup. So pause the video and spend a little time doing that. All right, let's take a look. It's a little bit messy. But at the end of the day, I'm hoping you got the answer had to look. I hope you got the answer, yes. It holds more than one liter, but just barely. So let's take a look. 

All right, first things first, I've solved for the height of the smaller cone up here, but I'm going to label it sort of down here. So what do I have? I've got this is now 22.5, right? And this now is 18 plus 22.5. Which is 40.5. Again, keep in mind, right? Just making sure, yeah, 40.5. You've got to kind of do that adjustment. But now you have everything. You've got the height of the large cone, the radius of the large cone, the height of the small cone, the radius of the small cone. So let's just lay it out. The volume of the large is going to be one third. Pi R squared times H, which is one third times pi times three. Whoops, nope, sorry, I'm doing the large, not the small. Get rid of that times 5.4 squared times its height, 40.5. All right, so the volume of the large will be this messy calculation. And I think we'll hold off from going over to the calculator and kind of working it all out. You should be checking your own work here, but the volume of the large cone is 1230 6.7. 1200 and 36.7 cubic centimeters. All right, simple enough. 

Let me substitute all the numbers in for the volume of the small cone, and then we'll just sort of write that down. The volume of the small cone, again, one third times pi times R squared times H gives me one third times pi, now we've got the three. Squared times 22.5. And again, messy calculation, be careful on your calculator, et cetera, et cetera. And the volume of the small cone worked all the way out, 212.1. Cubic centimeters. So finally, the volume of the cup, which is the volume of the large cone minus the volume of the small cone. 1236.7 -212 .1 is 1000 and 20 four .6. Yeah, or 1025 cubic centimeters. So. More. Than one liter. All right? Simple enough. Let me kind of pause it there. Check my answer. You go ahead and look at yours. We got it. All right, isn't that kind of cool? Now, by the way, lest you think I would never have to do this on a test. For those of you in New York State, the first common core geometry test had this exact problem on it. You had to use similarity to set up similar triangles, figure out the height of that smaller cone, do the subtraction, et cetera, et cetera. So let's do one more of these without any of the kind of help around it. 

Here we go. Bare bones, right? I go into my kitchen, I take out a cup. I measure probably the diameter is not the radii, right? Think about that. It's probably easier to measure the diameter of a cup than the radius of it, because specifically when you're looking at a cup, you might have some question about where exactly is the center versus the diameter, it's a little bit kind of easier to do. All right? But here we have it. 8 centimeter diameter, 5.6 centimeter diameter, a height of 15 centimeters, pause the video now, take some time and try to figure out what the volume of this cup is. All right, let's work through it. I have my ruler here only because it's kind of convenient if you've got a ruler in this scenario to just basically draw out kind of almost the other cone that you're really kind of trying to think about. So let me get that really quickly sketched on. There we go. I guess that's good. Right, again, it's like my ghost cone. This thing down here that I've got to deal with. Likewise, I might, since I have a ruler here. I might try to do it neatly. And. We try to find a better location for my center, but there we go. 

Toss this away, go over like this, erase that dot. And now I sort of have those right triangles in there, right? So what do I not know? I don't know this length, okay? So I'm going to call this length X that means my total length would be X plus 15. All right? I buy ruler. All right, so now I can set up a similarity to figure out what X is. I can actually use the 8 and the 5.6, but eventually I'm going to need the radii anyway. So this radius is of course going to be four. This radius is a little bit trickier, but half of 5.6 is apparently 2.8 with a big red 8. All right, let's try that again. 2.8. There we go. So now I have my radii. My four and my 2.8. Let's set up a ratio. I'm going to do four is to 2.8. Four is to 2.8 as X plus 15 is to X, right? This side divided by this side has to be the same as this side divided by that side. Let's do a little bit of cross multiply. All right, that's going to be four X equals 2.8 X plus 2.8 times 15. Now I want to get when you do 2.8 times 15, you get 42. 

All right, so again, kind of sparing us a little bit of time and going over to the calculator. 2.8 times 15 is that 42. I don't even want to do that in my head. Anyway, I'm going to subtract a 2.8 X from both sides. And what I'll get is I'll get a 1.2 X is equal to 42. Divide both sides by 1.2. And X is equal to 35. Oh, that comes out nicely. All right, 35. So 35 is now the height of my small cone. Maybe I'll even write that down right here. The height of my larger cone is 50. These numbers work out very nicely. 50 and 35. And now I have absolutely everything I need to figure out the volume of the cup. I just have to go through two fairly lengthy volume calculations for the cones. Subtract, and I have my final answer. So let's take a look at that one more time. The volume of the larger cone is going to be one third pi R squared times H, which is one third times pi times four squared times 50. All right, and again, let me just take a look, see what I've got there. 837.8, 837.8. All right, and what is that? Those are cubic centimeters again. That's the volume of my larger code. 

Okay. Let's take a look at the volume of my smaller cone. Volume of the smaller cone is going to be one third times pi R squared H one third pi. This one's a little bit messier. 2.8 squared times 35. All right, and the volume of that smaller cone. Again, make sure all the numbers are correct. Two 87.4. Cubic centimeters. Notice how I'm taking out these answers one decimal place farther than what I want the final answer near as cubic centimeters. So I'm rounding these other ones to the nearest tenth. All right, you could even take them out further than that, but don't take them out just to the nearest. I can make you off a little bit. So the volume of the cup is 837.8 minus 287.4 out of no red dot. 

All right, and that gives me 550. Cubic centimeters rounded to the nearest cubic centimeter. Yeah. And that's it. Literally. That's it. Right? And this is a wonderful modeling problem, because it really combines two very separate things that we did in geometry, right? It combines similarity in order to find a missing dimension that we need, right? And then it basically combines the idea of volume measurement for cones. Okay? And it's kind of a cool combination in a much, much larger problem, but that's often the most useful math. Is that when you can pull on a variety of different branches in a field like similarity and measurement and put them together to actually calculate something that's fairly useful. All right? So let's wrap all this up. 

What we did today was really look at a case study in measurement and modeling, specifically the modeling of the volume of a cup, right? And the cool thing about modeling the volume of a cup is it gets into those ideas of the whole being the sum of its parts, find an overall volume, subtract off a piece that's not there to find the volume that's left. And it also pulled on the idea of similarity, using the fact that a cone is being generated by a right triangle, and then using similar right triangles to find that missing measurement, the height of the smaller cone. All right. And that wraps up our final lesson in common core geometry. I want to thank you for joining us for these videos, especially if you've been watching them all along. All right, we'll probably add some more here and there as the topics arise. But for now, this is the last common core geometry video. So thank you for joining us until next time, remember my name is Kirk weiler. Keep thinking. And keep solving problems.