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Area of combination of solids

Let's learn how to find the surface area of combination of solids by breaking them down into cuboids, cylinders, cones, and hemispheres with the help of an example. Created by Aanand Srinivas.

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Video transcript

to understand finding surface area of combination of solids let's begin with a pencil and let's see where we go now if I ask you can you find the total surface area of this pencil how will you start thinking about it the way I'll think has hit is there a formula directly that gives me area of a pencil surface area of a pencil and it turns out that there isn't a pretty-looking formula so then I'll think okay can I imagine this pencil to be made of some shapes for which I know the formula in some previous year they taught me some formula can I imagine this pencil to be made up of those shapes and I want you to stop and notice if you can catch them in this case I've colored them so it's pretty clear the first figure you may see is a cylinder over here you may then see a cone over here and then you may see half a ball a ball is a sphere half a ball over here which is called a hemisphere so if you remember the formula for these or formulas or I just forget which of the two that is if you remember that then you can just find the area here find the area of this and find the area of this and add them and you will have the answer so I want you to pause right now and try doing that also check if you remember those formula if you don't we'll recap them over here the first thing that pops out to me is this cylinder so it's a three-dimensional object at the cylinder two circles if I cut it over here I'll get a circle over here another circle over here and I'll get this cylinder I'm going to try and draw that [Music] and what is the formula for this part of the area for this part like without the circles so if you remember the formula with the circles that's called total surface area I actually never remember that because that's usually not very useful especially in combinations I only remember this bits area which is called usually the curved surface area do you remember what that is that's right it's 2 pi RH 2 pi RH and a good question now is what are these R and H in this figure and the answer to that is R is the radius of this circle this circles radius is what we call our I want to mark it over here so that's this R and H is this height of the cylinder it's called height because usually we keep the cylinder upright and now I've made it lie down over here so this is my H so if you know these two you can find the surface area of this if you ever need to find the surface area totally just remember this and then add PI R squared for this and PI R squared for this at least that's what I do and notice that this curve surface area is just the circumference 2 PI R multiplied by H now let's go to the cone do you remember the formula for cone like as you recollect that let me try and draw the cone over here so I have my cone again there's a circle that I've cut out over here and now what is the formula for cone so again I only care about the curved surface formula only about this and I always only care about that I do not care about this bit because imagine if you were to paint this pencil right that's kind of what we mean by total surface area if you were to paint this how much area would you have to paint and notice that you won't have to paint this red bit because that's what's been cut that would have been inside so with this what is the formula for the curved surface area of a cone so it's PI RL by RL by the way you don't have to mark these two formulae up you can there is a way to understand why they work and I'll cover that in a different video but as of now it's PI RL and what is R and L R again is the is of this circle and L is not the hide though l is the what's called a slant height this length what's given directly for Isaiah we are finally left with this guy this half a ball or half a sphere which is also called a hemisphere we look something like this let's draw it there it is and I have something that looks like a football that should do and you need to find this area notice it it's a 3d figure right I mean I can't draw 3d on this so what is this area gonna be do you remember now if this is R if this radius of this circle that you've cut here if that's our and once again if you care only about this curved surface because you can just add PI R squared to it then this curved surface area is actually a very very interesting number it's 2 PI R squared now why is that interesting it's because this circle would have been PI R square but this area is 2 PI R squared and notice that if you remember these three actually you can solve all the problems that are asked at least in class 10 these are the only three formulae apart from of course length into breadth for rectangle and PI R squared for circle that's all you need to know so now's a good time to actually use the numbers in these results and see what the final answer is I'm gonna do it now let's see if you get the same answer so pi RL that's 22 by 7 that's usually what I take bias unless they say something else x r r is 2 centimeters that's a mistake that I've made many many many times it's actually R is 1/2 of 2 centimeters because this 2 would be the diameter of the circle it's the width of the pencil or the diameter of the circle so R would be half of that our would be 1 centimeter 1 centimeter so 1 centimeter multiplied by L which is directly given to us as 2 centimeters that's going to be equal to 44 it's right over here 44 22 times 2 by 7 centimeters square centimeter into semi the reason I keep the units here is because I used to always like Nazi say something will be given in millimeter sneakily and then I'll not notice that and I don't make a mistake so I try my best to keep the units so that I'm in you my's the chances of going wrong I still go wrong the number of times I mean calculation errors and this is very high so what about two pirates two times 22 by 7 x r which is 1 centimeter notice that the R is going to be the same the circle and the circle are the same for oh here and here 1 multiplied by H it is this length right that's 10 centimeters let's draw a line again over here just to keep things separate now let's find this this is gonna be 2 times 22 44 times 10 440 so 440 by 7 centimeters square shouldn't surprises clearly the surface area should be much bigger than this it turns out it's 10 times more 10 times bigger now you have this hemisphere over here that's 2 PI R squared so you're gonna take 2 times 22 over 7 multiplied by R squared R is 1 so 1 centimeter Square and this is also gonna be 44 by 7 that's a coincidence in general this and this need not have been equal at all so now we have it now let's look at this what is the total area of this going to be just the sum of these three 44 + 4 4 t + 44 in other words 440 + 88 I'm gonna do that in my head 440 + 88 that's gonna be four hundred 40 plus 100 minus 12 so 545 3528 so 528 over seven centimeter square and you can leave it here or calculate what this will be I'm gonna just do that so seven goes seven times and 49 and I'll have three remaining and that's 38 it goes something goes five times and 38 and then I'll have three more remaining and in three in 30 it'll go four times and I'll have remainder of two you just go under three times I'm gonna leave that the surrounded off answer and one of the problems I had at least when I used to solve such questions is whenever I look at an answer like this I would be like I'm deaf I've definitely made a mistake because I I'm used to getting pretty looking answers but don't worry in this chapter you do get a lot of answers that look something like this and finally the purpose of this video is not to find the surface area of a pencil even though that's what we did it is to use this question as an excuse to learn how to solve any problem in this chapter because all the problems follow the same pattern look at a combination like this break it down into its pieces in this case a cone a cylinder and a hemisphere it may be something else in another question but what you'll be doing is exactly the same thing then remember the formulae that you need to solve the problem I actually chose the pencil here because it gives us neatly the only three formulae we need to remember which is PI RL for the curved surface area or 2 pi RH for the cylinder surface area and 2 PI R squared for hemisphere you can solve any problem just using these three as long as you also know a rectangle which is a length into breadth and PI R square for circle once you've done this it boils down to checking the numbers correctly and putting them in without making any mistake and you will have your answer