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Volume of combination of solids

Let's learn how to find the volume of a complicated solid by breaking it down into cylinders, cones, hemispheres, and cuboids. Created by Aanand Srinivas.

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Video transcript

let's learn how to find the volume of solids that don't look like the typical cylinder or cone or a cuboid no things that we've already learnt let's look at something that's slightly more complicated than that what if we had to find the volume of something like this say a pencil here are some lengths what methods can you think of of finding the volume of this pencil now one method that strikes me immediately is hey I need to find volume right but volume is just the amount of substance that's contained in this so what I can do is take this and immerse it basically dip it in water and look at the amount of water that gets displaced you know it's basically pours out if the container was completely full the good thing Archimedes tells us is that the volume of that water that spills out and this volume volume of the pencil that we want will be exactly the same and then I can what I can do is take that water and what does the good thing about water it has no shape it will take whatever shape but I wanted to take pulled in some beaker and I can measure the volume that's probably what I would do if I really had to do this in some for some reason the other method because this pencil has this beautiful property that it looks like hey there's a cone here that I can see there's a cylinder I can see here and there's a half ball or a hemisphere I can see because this pencil looks happens to have these things within it we can also apply this other method where I can break this pencil down into a cone a cylinder and a hemisphere and this is the kind of problem we will see in this chapter solids which we can break down into things that are familiar to us and just add and the good thing about volume is that you always just have to add the amount of substance just adds so you can pause the video right now and break it down find each of the volumes individually and add I'm going to start so the first thing I have is a cylinder is Lynda so a cylinder has some radius like this if I were to cut this out over here and then it'll look something like this obviously it'll be longer I'm just showing it's small over here and then this will be the circular part this will be the height do you remember the formula for volume of a cylinder it's PI R square H i R square H and you can ask me okay is that an intuition that helps me remember this better the way I remember it is the area of the circle is PI R squared you're stacking this PI R squared H times because this is H and this area is is R I'm sorry this length is R so this circle the area is PI R squared and H times you keep it you get PI R Squared's now for the cone if I were to draw the cone over here so there is my cone and once again oh my god that circles probably the worst one I've drawn so this looks better yeah this cone what is the volume of this cone the formula for it again you don't have to remember it because as you solve more problems just keep looking it up in the book and eventually you will not be able to forget it so it's 1 by 3 PI R square it's now one way I remember it is to notice that it's 1/3 of a cylinder of the same height and the same radius right if I had a cylinder of radius R and a height H so in other words if I had a silicon sorry a cone like this that'll be one-third of the area of that cylinder that's just a way to remember it the way to actually derive this and to see why it works sort of prove it happens with me in class 12 if you learn integration you can derive these quantities using integration the third thing we have your is half a ball which is like a hemisphere so let's let's draw the hemisphere over here I'd say we again have a circle and we have a hemisphere there it is and the volume of a hemisphere the more popular formula remember is the volume of an entire sphere so volume of a sphere is actually 4 by 3 PI R cubed 4 by 3 PI R cubed so the volume of a hemisphere will be 2 by 3 PI R cube half of that and the good thing about having a volume is that you actually don't have this concept of total volume surface volume and all that volume is just the amount of matter no matter what you do it's the it's the same thing it's just the amount of matter so you just have to add it these are the only formula you have to remember for volume apart from of course Q now the cuboid volume is actually very intuitive its length into breadth into height now why that's intuitive is because very similar argument to this you take a rectangle of length into breadth area and then multiply it keep it stuck at each times so L into B into H very similar to PI R square H now the question boils down to just finding the right numbers for each of these and adding them up you can go ahead and do it I'm gonna do it now so it's 1 by 3 into PI I take this 22 by 7 into R squared now once again R is not 2 centimeters here it's the width like we saw on surface area you have to make sure that you take half of this so it's one centimeter and I'm actually going to move this a little bit to the left let's move this so I have space to do cylinder and there it is so R is one centimeter and you have R squared so one centimeter actually put one to one centimeter squared the reason I keep the unit's here is to not make mistakes it's over here is for the cone what is it it's two centimeters given over here so it's directly given to centimeters notice that for the surface area you actually care about this length the slant height but for the volume you care about the height directly in case you wanted the slant height you can actually find it using Pythagoras theorem this is 2 centimeters this is one centimeters so you can find this root of 2 square plus 1 square which is root 5 but that's not needed for us that was actually why did I even say it I don't think I should have said it it's not necessary but yeah what about a cylinder over here this does cylinder over here so you have 22 by 7 let's use blue 22 by 7 multiplied by R squared and as once again just 1 centimeter into 1 centimeter square multiplied by 8 which is 10 centimeters 10 centimeters so we can calculate this later what about the hemisphere that's going to be equal to 2 by 3 2 over 3 into 22 by 7 into R cubed an R in this case is 1 centimeter cube notice that in all these cases you finally get an answer that's in centimeter cube which is basically cubic centimeters it's called it's the unit for volume and this also helps you check that your units are fine so let's do it now what do I have I have 22 into 244 divided by 21 centimeter cube over here I owe here I have 220 by 7 220 let's use blue 220 by 7 centimeter cube and over here I have 44 by 21 again 44 by 21 coincidentally the same volume as this part of the pencil coincidentally so there you have it you can actually find the total volume you just have to add this and this and this the answer may or may not look pretty so you can go ahead and do it 44 plus if I just multiply this with a 3 and a 3 over here I will be able to add them all because they have the same denominator so at 660 plus 88 660 plus 100 760 minus 12 750 748 so 748 divided by 21 centimeter cube I'm gonna leave this one here because I know you can find what this number should should be and once again notice that the purpose of this video is to not just find the volume of this pencils but a notice that this will be the type or this will be the pattern that will happen that that you'll follow for all the questions regarding volume of solids like this you will first break it down into pieces that for which you know the formula and then you'll remember the formulae that you need to remember actually these are the only things you need to remember for a cone it's 1 by 3 PI R square H for a cylinder it's PI R square H for a sphere it's 4 by 3 PI R cube which means for half a sphere it's 2 by 3 PI R cube apart from this like I said the only other one is a cuboid which is length into breadth into which L in to blend into breadth into height that's it there's nothing new that you need to know after this that you need to solve problems in this chapter