19 2 Surface Area of Prisms and Cylinders

hey everybody it's your old buddy mr evans here we're gonna be talking about surface area of prisms and cylinders this is section 19.2 i'd get out piece of paper just to work through some of these examples and have calculator handy all right so surface area pretty straightforward meaning it's telling you exactly what it wants you to do you're finding the total area of all the two-dimensional faces or surfaces of the figure all right of three-dimensional figure so we're actually only going to use the formula for couple of the shapes the cone and the sphere otherwise we're just going to be finding the areas of all those shapes that make up the surfaces of the figure and add them all up all right so for this first one be pretty simple always like to label my work and be kind of organized with this so like the top and the bottom face they're the same right how do find the area of that top and the bottom well this is rectangular prism so how do find the area of that rectangle 10 times 4 right length times width so 10 times 4 is 40. i'm going to go ahead and multiply that by 2 because there's two of them right there's one on the bottom as well maybe color it in just for funsies all right then maybe we do the front and the back obviously the order doesn't matter right front and the back so now we're looking at the dimensions 10 and 8 right to look at that front and the back so we'll go 10 times 8 which is 80. obviously there's back side as well that's the same size so we multiply by two we get 160. so all that's remaining is the left and the right sides all right those dimensions are four by eight to get the area of that so we're gonna go four times eight which is 32 32 times two is 64. and then we're going to go ahead and add all those up and we'll have the total surface area of our rectangular prism all right 304 square inches all right so we may have little project at the end of the year where we're having to find you know how much paint we need to use for room or something obviously that would be fine in the areas of the walls of the room right so the the formula if you want to use it you can go ahead and look in the textbook really just think it's waste of time though think it's way easier just to find the area of those faces so why don't you go ahead and try number one on your own go ahead and pause the video all right so here we go work for number one again we're going to try to be as prepared as possible so top and bottom our dimensions are eight times four or check that eight times let's say six eight times six 48 48 times 2 96 that takes care of the top and the bottom maybe we go left and right now so i'm looking at this right side those dimensions are 12 by 6. so we're going to go 12 times 6 which is 72 72 times 2 is 144 and it looks like we're just missing the front and the back all right those dimensions for the front see 8 and then that will be by 12 8 by 12 to get the area of that front there so 8 times 12 is 96 96 times 2 is 192. then we're going to add all those up and we got our total surface area 432 square centimeters all right okay so here we go number two if you want to try it on your own go for it otherwise i'm going to go ahead and do it so we have triangular prism all right so this is where we just obviously we need to know how to find the areas of triangles and find the areas of rectangles all right so let's just start with this top triangle so we're going to look at two different triangular prisms number two and number four and there's going to be reason why because number two is right triangular prism number four is an equilateral triangular prism and just want to go over the easiest way to find the areas of those all right so if want to find the area of this top the top and the bottom in this case remember right triangle we have the two measurements already the 10 and the 24 right so remember the base and the height of triangle are always perpendicular to each other so in this case the top and the bottom would be 10 times 24 over 2. now some of you may figure this out if remember it's base times height divided by 2 for triangle but there are two of those so really we're going to be going 10 times 24 divided by 2. that's 120. that's for the top and then i'm going to go ahead and multiply it by 2 again because there's two of them so either way you want to think about it that's kind of the long way we could have just gotten 10 times 24 right because we divided by 2 because it's triangle but then there's two of them anyways the top and the bottom would be 240 all right so let's say want to look at so now if you think about it these lat with what's called lateral faces so we have this area and we have this bottom area now so now we have three lateral faces so we can't just it's not like rectangular prism where i'm just gonna take one and multiply by two we have to find the area of each rectangle separately all right so let's say we wanna look at this this backward let me erase the top and the bottom here so if want to find the area of this rectangle back here pretty simple right 24 times 20 to get that little fella right there so let's go ahead and handle that 24 times 20. believe that's going to be 480. all right and then we'll find the area of this one on the side over here so that looks like it's going to be 10 by 20. there's my 20 right there 10 times 20. which is 200 so here's our here's little issue here is if want to find that the area of this front one notice we don't have this dimension we do have the 20 but we don't have that one so we will have to use pythagorean theorem on this top triangle to get that length all right so 10 squared plus 24 squared equals squared that is version of pythagorean triple and i'll just go ahead and save us the work it is 26 inches but just be aware of that when you're using these right triangles you may have to use pythagorean theorem for the right triangle so again for the area of this front one now now we can just go 26 times 20 and we're on our way believe that's 520 yes nailed it so here are four faces right in triangular prism that should make sense really are five faces right the top and the bottom triangles and then the three lateral faces add all those up and we should get 1440 square inches all right so if you want to go ahead and try number three go ahead and pause it go ahead and pause it now i'm just going to put the answer in there in minute so the answer to number three rectangular prism should be 410. square inches all right we're going to go over one more you guys real quick because that's all you're going to see today actually we've got cylinders too but anyways i'm going to go over one more prism and then we'll do cylinders really quick all right we'll do composite shapes tomorrow so here we go equilateral triangles remember we have that special formula we can use 1 4 root 3 times side squared all right so for our front and back bases here we're going to use that all right pretty simple here so let's go for the front and back we'll call it 1 4 root 3 times 6 squared so in my calculator i'm going to go let's go square root of 3 first we're going to get decimal answer so that should be about 1.73 times 36 divided by 4. so for the area of one of those triangles we should get about 15 rounding it would be about 15.6 but i'm not going to round it yet because remember that's just one of those triangles let's go ahead and multiply that by two and then maybe i'll round it now so i'm going to use 31.17 that's for the front and the back all right we got those handled now looking at this since this is an equilateral triangle all three of the rectangles that make up the lateral faces so the right side the left side and the bottom they're all the same size they're 6 by 12 rectangle all right so 6 by 12. so we could go ahead and just maybe call these the lateral faces we're going to go 6 times 12 and then there's three of them so 216. add all that up we get about 247.17 so it's up to you if you want to just go ahead and handle all the the prisms now in your homework or you can go ahead and take look at surface area of cylinders all right so surface area cylinders we will use the formula if you think about here's the net of cylinder right here on the right if you unroll the lateral part of that it just ends up being rectangle all right and so basically what we're doing is we're finding the area of the two circular bases and then we're going to add this area this rectangle and you can see that the dimensions of the rectangle are 2 pi the circumference of the base ends up being the length of that rectangle and then the width is just the height of the prism mean in reality follow this formula everybody because there's nothing can do about you guys using your notes on the test right so is our radius is our height so let's just work through these really quick you can leave your answer in terms of pi so we're going to go 2 pi 3 squared plus 2 pi okay we should get 18 pi again doing exponents first that should be what 54 pi and then add them up so our total surface area for number five 72 pi centimeters squared notice when add up these pi's 18 pi plus 54 pi just like using any other variable just add up those numbers we get 72 pi score centimeters and think that should be good you could probably go ahead and stop the video now and start your homework if you're interested in seeing another cylinder go ahead and keep watching do that right now all right so 2 pi 6 squared plus 2 pi 6 times 15. so that's 72 pi that's plus plus 180 pi so we get it's at 252 pi centimeters squared all right everybody good luck have great day talk to you later