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Problem types: surface area of combination of solids

Let's look at a few common types of problems involving 'surface area of a combination of solids' and look at broad strategies on how to solve them and what mistakes to watch out for. Created by Aanand Srinivas.

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Video transcript

so what are the different types of problems you can be asked in combination surface area of combination of figures these are the broad three types in my opinion the first one is straightforward you have some cylinder some cone both are next to each other they have the same radius find the curves of his area find the curved surface area over here add them if you're asked to include the bottom add another circle if not don't include it similar story here everything has the same radius rady all of them have the same radii and you find the curves of his area curves of his area cylinder add them this is the most straightforward there's a small twist he can add to it in type two type twos where see if you notice over here this question this is like something something like a tent right and this one like an arrow what's the difference between the two if you had to find the total surface area in other words if you had to paint this on the outside and if you had to paint this what should you be careful about that's right you should be careful about this region over here let's actually draw that out so if you look at it from the bottom if you're looking at it like us little ant who's under the sand is like looking at the whole thing and then what would you see you would see two circles right a smaller circle for this one and then a bigger circle for this one now notice that these two don't have the same radii because if you drawn the same thing over here you would have seen one circle for this one but another circle of the same radius for the top one so you might see something like this but over here you seeing two different ones and which of these two areas do you need to care about is it the smaller circle well if you if you're counting the bottom then yes but then even after doing that if you just care about this region you have to add that region as well and that's going to be this area out here now again nothing new has to be learnt to do this you just have to find the area of this circle bigger circle and then subtract the smaller circles from it so bigger circle minus smaller circle that's a small difference away a similar story over here so the second type is where the radii don't match other sizes don't match there are these gaps over here so how would the top view or the bird's-eye view of this thing look this looks something like a building with a little dome on top of it it's a bit of bird flying about trying about what would you see you would see this square four square a rectangle whatever that is for a box and you would see a circle right this dome seen from the top is actually a circle of this radius so you would see this and if you do decide to paint this place you care about the curves of his area of this you care about all these rectangles a cuboid is just a bunch of rectangles you care about all of them but you also care about this gap region and how do you find that gap region once again you care about this region on the outside so you would find the area of this box and then subtract this circle and that would give you the area that you need a lot of the skill in this chapter is just being able to visualize these 3d shapes and make sure you're not missing anything another twist type number three I call it is where you are given some kind of like say a pen old holder over here right where there's this cone thing that has been removed and here your skill is to be able to identify what's not there seeing what's not there looking at this figure and going okay if I had a cuboid and if I had removed a cone from it then I can find the surface area of this if I imagine this to be a cuboid with the cone removed that'll give me the surface area and the intuition or the other immediate thought is that okay here i should add the surface area of this hemisphere or cone but here i should subtract its pause for a moment and think of that is true because that's a common thing that I see at least when I have done earlier so let's do for volume if you're caring about the amount of substance in here then you care about how much you remove if you remove this cone you've removed some volume but if it's surface area if you're thinking about painting it does it matter whether you're painting a cone that standing or a corn that slick inside the figure you have to paint it anyway right so actually these two cases right where in one case the radius is similar but it's removed that's this is more similar to something like like this case and over here where the radii are different this is actually smaller than this cuboid but it's again removed is not that different the main difference is that you have to treat it similar to this problem and let me tell you what I mean by that if I have so how's this the top view of this look or the bird's-eye view you would have a box and you would have this cone as just as a circle right and just like over here that's why I said it's similar to this case you care about this area you care about all the other areas just like over here say if this had been a cone and finally you will still add the curved surface area of the cone not subtracted because you have to go inside and paint like you would see some point over here that represents you know the end of that cone the tip of that cone just like you would do over here if you had if the cone had been over so the cone being above and the cone being below does not make a difference to your answer that's actually a little counterintuitive at least to me so you can draw the top view of this if you want to so the main purpose here and I call this the problem space of this chapter these are the only types of problems you have to care about the moment you see it you can say okay is this the same radii kind of question okay though or is a different one or maybe it's going inside but then you know what to do in each of these cases