Finding the Composite Surface Area of 3 D Objects

okay so what we want to do is basically go through how to find the surface area of composite 3d object so when two different objects are stacked together or maybe they're made from two different objects and we're trying to find the total surface area so the big thing we have to pay attention to here the new piece of information that we need to be thinking about is that overlap where these two objects are stacked because where it overlaps is no longer part of the surface area so there's really no new math here but these problems can get pretty long in the solving you just have to be really patient and kind of stick it out you will get that moment of whoo did it you know or it's done it just takes lot more time to get to that point then maybe some of the other math problems that we do so this is one of the longer types of math problems that we look at in grade 9 but there is no new math here that you don't already have the skills to do so we are going to be following our sort of instructions for finding the composit surface area so we're gonna follow step so our first step is to find the surface area of each individual shape that makes the compound object using the surface area formulas so guess that would mean need to determine what my two shapes are so in this case have cylinder here and have rectangular prism looking at the dimensions again so knowing that I'm actually just gonna go ahead and label my two objects just so know what I'm talking about so object two remember one is gonna be that cylinder object number two is going to be this box and we're going to start by doing that first step so we have object one okay and object one we need to find that formula so these are the same steps that we did in just finding regular surface area so I'm gonna find my formula sheet I'm going to find my formula for surface area which is right there and I'm just gonna go ahead and write that down so the surface area is equal to two PI squared plus two pi RH now when these formulas are made they include calculating for that bottom piece okay because of that there is way to combine your formulas and take out the bits of the formula that have to do with that overlap and and sort of shorten this process but honestly found that when students try and do that they just get more confused so we're gonna do it the straightforward way the way that found to be most successful with the most students over the years and that is just by finding the surface area of each object individually adding those together finding the overlap and then getting rid of it from that total so here we are with our surface area formula for the cylinder we're gonna go ahead and we're gonna write out our variables just like we always do so we're gonna use 3.14 and I'm gonna ask that you guys use 3.14 4 pi as well just to keep that consistency in your answers and that way it'll also match the answer key when you solve other problems I'm still pi is 3.14 our radius remember that radius is only half the distance across the circle so we have 6-foot diameter so our radius is half of that 3 feet and then we have our height and our height is 8 feet so we've got all of those dimensions written down I'm just gonna pause here for second and put those variables in and then we'll keep going ok so now that we've got those pieces in there so I've got all of my variables replaced with their values can go ahead and do those multiplied so I'm following order of operations just like before and we're going to go 2 times 3.14 times 3 squared which would be 9 and we should get 56 point 5 2 and then over here I'm going to go ahead and do that multiplication two times 3.14 times 3 times 8 which are just those variables replaced with their values and should get 150 point seven two okay so if go ahead then and do that addition between the two shouldn't have with total surface area for that cylinder of two hundred and seven point two four feet squared all right so found my surface area of object one but I'm not quite done step number one yet because need to find the surface area for object two so this is all just step one from those steps that gave you so object to surface area is that rectangular prism and if we check our formula sheet again we find it on there so we're gonna write that down we've got two times width height plus length width plus length height and we go ahead and write out all of our variables so remember that it doesn't really matter which dimension you use as your length your which and your height width and height as long as you're consistent so we're gonna say that our width is 10 feet our height is 7 feet and our length is 4 feet so I'm gonna again put those variables in there and then we'll keep going okay so if you don't remember what I'm doing well I've kind of paused here and putting those numbers in go back to that previous video about finding surface area and that should help you out so we're gonna go through our order of operations just like we did before you've still got 2 outside there sits outside the brackets 10 times 7 gives us 74 times 10 is gonna give us 40 and 4 times 7 it should give us 28 so there is our mounts and then we're gonna again go another step add those together and we should find hundred and 38 I'm always so nervous now if videoing this that mean to make that error and it'll be trapped forever on history of film and then we're gonna go ahead and finish that last step of multiplying outside the brackets to the inside of the brackets and you should get surface area for that box of 276 feet and again it's square because we're generally three-dimensional objects so in surface area so now what we're going to do is find the total surface area for both of the objects and that's going to be that step two okay so we talked about step one finding for each step two is to actually add those totals together so we're gonna take surface area of object one and we're gonna add it to service area object two if you had more objects stacked together you would add all of them together once you'd found them all so in this case if we add those to 207 0.24 and 276 together we should get four hundred and eighty three point two four feet squared okay then we're gonna go ahead and find the overlaps so step three I'm just thinking back to our steps is to find that overlap and it says here that we're going to multiply it by two and the reason that we're multiplying it by two is because the overlap is coming off the surface of the cylinder but it's also coming off the surface of the box where the cylinder is covering it so because of that we're gonna take that overlap area way twice so our first it has to be let's figure out what is the shape of the overlap so in this case my overlap so that's step three okay my overlap is circle so I'm going to go to the 2d side of my formula sheet and I'm gonna find the formula for the area of circle which is PI squared so I'm gonna go area equals PI squared not gonna worry about the multiplying by two yet and then I'm gonna write out my variables just like always do so pi is 3.14 was three so it's the same on the bottom as it was on the top and we're gonna go ahead and put that into our formula and if we do 3.14 times three squared we should find an area of 28 point to 6 feet squared okay so now that we have the area of the overlap we're gonna take that and we're going to multiply it two by two to account for the fact that it's on both of those objects so I'm gonna times that by two and my total overlap equals fifty six point five two feet 56.5 - there we go so we've got hurt over total overlap right we've got our total surface area right there and we've got our trouble right there so now we're ready to go ahead and do step four which was to subtract the total overlap from the total surface area so I'm going to go for eighty three point two four so this is our step four sorry feet squared and I'm gonna subtract fifty six point five two feet squared and that is gonna give me my total which should be four twenty six point seven two feet squared so there we have it there is our final total I'm just to show that that's our final answer but that is the composite surface area taking into account the overlap finding all the surfaces and getting rid of where it overlaps okay so hopefully that will help you along and we will go from there