Problem types: volume of combination of solids

our purpose here is to look at some types of problems that you can be asked in finding the volume of combination of figures and notice that all you need to remember to do them is just the old three formula let's go one by one one is a direct addition you're given some figure this looks like the thing I used in chemistry lab or at least I broke in chemistry lab I think it's called a round bottom flask so over here as you see this if you have to find the volume of this you will notice that you can imagine it to be made of a sphere a sphere over here and a cylinder you will approximate of course like you can imagine this to be as forgiving it's very small and then you have to add the two volumes and you'll get the final volume over here you can imagine it to be made of that's right one cylinder over here and two hemispheres similar to what you may have done in surface areas and that's the addition one the other type that I feel I mean it's not exactly another type in the sense that you use a different method but it is something that you might see that looks different from this where it's not added but it's removed so you're a hemisphere has been removed from a cylinder here a cone has been removed from a box or a cuboid and here maybe you're making a pen stand here and here maybe you're making I don't know pond or something and in this case you have to find the volume of the bigger figure and then remove the volume of the smaller and in this case you will actually subtract because it's the amount of substance so removing something here it actually means subtracting the volume you may remember that in surface area though even if you are removing this area will get added but in volume this will get removed so in some sense volume is a little bit more intuitive i removing a volume leads to reduction and then there is no concept of curved surface volume total surface volumes none of that volume is just the amount of substance so you don't even have to ask questions like hey should I include this bottom one or not none of that because volume is just the amount of substance like I just like I just said so the other type that I feel is once you have mastered these two you can do a little bit more with them it's not new but you'll be combining things so here for example you'd be asked what is this remaining area over here and I've drawn it in in a 2d way over here what I mean is let's say this is a cylinder and this is a cone and this is a hemisphere you put this combination inside and ask how much area a volume is left over here and the way to do it is to find the volume of the hemisphere find the volume of the cone add the two and then subtract that their entire thing from the volume of the cylinder and you can notice that this is a longer problem but doesn't mean you need to know anything and you know to do it another way to make the problem longer if not more difficult is to give you many such you take a body like this this is like um these are like small little versions of this and put many of them so now you have to find the volume of one and then multiply it by whatever number there is so these are all ways to make the problem slightly long and to do all of these the only ER formulae that you need to know are PI R square H for a cylinder 1 by 3 PI R square H for a cone basically 1/3 of a cylinder and 4 by 3 PI R cubed for a sphere so this is for a sphere this is for a cone and this is for your cylinder and of course I'm assuming that you are comfortable with length into breadth into height for the wall finding the volume of a box or a cuboid like this