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Volume of a frustum

Let's learn how to find the volume of a frustum by looking at an example problem. Let's use an intuitive approach to find the volume rather than directly applying the formula. Created by Aanand Srinivas.

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Video transcript

if you want to find the amount of water that a container that's shaped like this can hold if you know these lengths how do you how do you find the amount of water it can hold the amount of whatever this can hold is just another way of asking what's the volume and we particularly care about this shape like this is not one of the common shapes right this is not a cone this is not a cylinder but it's something in between a cone and a cylinder if you if you think about it because it is like converging like this but not completely all the way to become a cone we call this a frustum of a cone I like to just think of it as a bucket or a or a tumbler or an ice cream Cup and I think that we are specifically studied this shape because this is pretty common right a bucket is a common shape and the amount of water a bucket holds is something that you can actually care about in the real world so how do you do this you can go ahead and maybe like look up the formula somebody has derived it for this particular shape itself if you put in this length this length and this length but if you like me if you don't remember it or if you don't know it or if you don't like to remember it the way to do this is to do what we're doing for all the other problems which is look at this shape that's little unfamiliar and asked can I represent it as solids that are familiar to me and in this case the answer is yes you can just imagine it to be made of a bigger cone from which a smaller cone has been cut out so a large shape that's familiar for which we know the formula we spend some time remembering the formula for the volume of a cone right it's 1 by 3 PI R squared H so we remember that so find the volume of this larger cone subtract the volume of the smaller cone and you'll get the volume of this now that's the strategy you can go ahead and do it except that there are some lengths missing right if you notice I'm just going to label this now if you notice to do that I need the height of the smaller cone and I need the height of this bigger cone both of which have not been given to me directly now I see directly because they have been given to me indirectly like the moment I see this length this length and this this height given to me there is only one bucket I can have with these dimensions which means there is only one I can draw that completes them but how do I find this height h1 the observation I make is this triangle over here I know this is a cone but if you look at it right from front in other words you look at a cross-section this will be a triangle this will be a triangle and they will be similar triangles this angle is equal and this is 90 this is 90 so a a similarity if you want to be precise about it then what we can do is write this side by this side will be equal to this side by this side the reason this should strike me is that I'm looking for a relationship such that which I can use some equation that I can form that will connect my unknowns my h1 to mine owns over here so now I can actually go ahead and write that equation what would it be h1 by 6 or I can I can write it the other way as well I can write H 1 by H 2 will be 6 by 9 that's another way of another way of writing it so I'm going to write that H 1 by H 2 equals so my unknowns on one side and I have my knowns on the other side 6 by 9 now a good thing is H 2 is not independent of H 1 right it's just H 1 plus 4 so I can go ahead and write H 2 as H 1 plus 4 which means I will reach that state that I like which is one equation one unknown I know that solvable so I can just cross multiply and find the answer so it's going to be 9 H 1 equals 6 H 1 plus 4 into 6 so 6 H 1 plus 6 into 4 that is 24 now I can subtract 6 H 1 on both sides that will give me 3 H 1 on this side and 24 on this side which finally gives me H 1 equals 8 centimeters notice that this is the key idea the key idea is being able to visualize this cone over here using similarity to find this h1 over here with which you can find h2 it's two equals this is 8 this is 4 so this is 8 plus 4 or 12 centimeters and you've done all of the hard work you need to solve this problem because now it's just about writing the volume of this bigger code which is 1 by 3 by r2 square I'm going to call this large radius r2 r2 squared H 2 that is the volume of this big cone I want to subtract from this the volume of the smaller cone which is 1 by 3 PI R 1 square I'm going to call this small radius R 1 square H 1 now notice that you have all of this our tools available H 2 you found our ones available and H 1 you found which means now this boils down to doing the calculation without making any mistake now let's do that so 1 by 3 into PI can be taken out common 1 by 3 into PI I'll take it as 22 over 7 let's maybe draw a line here so that we know that this is where we did that calculation this is where we're finding the volume and this can be multiplied by r2 squared which is 9 squared 9 squared x I'm gonna write 81 for 9 squared 81 into H 2 which is 12 81 into 12 that looks like 81 point well minus r1 squared which is 6 square that's 36 36 into H 1 which is 8 centimeters into 8 now you can notice everything here is a number whatever you get in centimeters cubed will be the answer now you can go ahead and do this calculation maybe you can take something out common I'm I just did the calculation and what I got was a seven one one second let me just check seven one six point seven one six point five seven seven one six point five seven centimeters cubed which is another word of saying about 716 ml centimeter cube is the same as milliliters or 0.72 liters so it's under a liter about 750 ml is the volume that we got for this so it's not really a bucket it's probably a tumbler a large one so that's what we have here now what I want you to know is that the only way I can make a problem like this difficult is by giving you numbers that become really messy some couple of decimal zero decimals you do all of that so that this calculation here itself will take you many minutes but notice that this part if you did have a calculator would become very easy so the method as it is is pretty straightforward imagine it to be a big cone - a small cone find the height of the smaller cone using similarity like we did over here use that to find the height of the bigger cone after which you can just subtract the two volumes so I used to find it pretty weird when these numbers got really large so don't question the method you are using because the numbers get large because that's what that's what I was doing at least when I had to solve these questions so with all that any question you see it's better to use this in my opinion than to have to plug into the formula which does not provide you any intuition