N Gen Math 6 Unit 10 Lesson 3 Surface Area

hello and welcome to another engine math 6 lesson by emathinstruction my name is Kirk Weiler and today we're going to be doing unit 10 lesson 3 on surface area so this entire unit is about solids right and we normally don't think too much about the area of solids that doesn't actually make lot of sense because area is the amount of two-dimensional space that two-dimensional object takes up so what do we mean exactly by surface area well we're gonna get into that in the first exercise so don't want to spoil it right now that wouldn't be any fun let's start by taking look at the first extras and now all solids we have studied so far have faces that are polygons right whether those faces are triangles or parallelograms or whatnot but they're all polygons right and each one of those faces has its own area because all the faces are two-dimensional objects and that gives us little bit of hint about what surface area is gonna be but let's dive into exercise number one and talk about it right triangular prism is shown below in two different views letter how many faces does this polyhedron have alright polyhedron is just fancy word for like solid ok that's enclosed by polygons all right at the end of the day you don't need to know what polyhedron is but this thing is is imprisonment alright so how many faces does it have pause the video now and quickly count that alright well it's simple enough right we've got one two triangles and then one two three other lateral faces so we've got five total faces simple enough make sure that that looks like five and not six there we go little bit better alright now let her be important how do you know that the lateral faces of this polyhedron are rectangles all right so we know that the two bases are triangles because it says it's triangular prism but all of the other faces the lateral ones that you really kind of think about more in this sort of view right because now here it's actually lying on one of its triangular basis how do we know all the lateral faces are rectangles what what tells us that pause the video now well what tells us that is that it's right prism all right well the fact that it is right prism all right whoops anytime you're told that particular prism is right prism right then that really kind of means technically that the height is perpendicular to the base right and because it's perpendicular base all the heights are perpendicular the base which makes all of these angles right angles and all of those lateral faces that doesn't look like right angle but there we go all of those lateral faces are now rectangles now it's little bit tricky because the two triangles as indicated on the picture are actually right triangles all by themselves so it may seem like this name is basically saying we got our right triangle as its base and then we got prison but right triangular prism is simply prism whose bases are triangular and then whose lateral faces are rectangles now let's take look at letter very important draw each of the faces below and show the dimensions of each all right so this is easy this is literally like if put up piece of paper and kind of traced out this face and then kind of put it down so that was just two-dimensional right what would it look like well it would be right triangle right that would be six centimeters tall and eight why right now keep in mind that this is one of the bases and so because that's base there's an identical face to it right so you're gonna draw that as well right 6 & 8 think I'll leave off the centimeters for right now we can see them on the picture and now right those are two of the five faces I've now got three rectangles so what are those rectangles look like well mean one they look like rectangles one of them one of them has length of 10 centimeters and guess height of 2 centimeters so let me draw that one right we've got this one that's 10 by 2 all right we've got another one that is 6 centimeters wide and 2 centimeters high so that's sort of this rectangle if you will and on this picture up here it's this one back there all right let me kind of maybe bring this up here so we've got one that is 2 by 6 and then finally we've got the one that's kind of back here right and that one is 8 centimeters this way and again 2 centimeters that way because it's the same as this particular length so finally we have 8 & 2 so we've got these five faces right three lateral faces and two identical bases now letter no great surprise here we're gonna find the area of each face and we're gonna sum them all up and that sum that sum is known as the surface area of the solid right so that's all surface area is by the way surface area is simply the sum of the area of all the faces that enclose solid and it makes lot of sense right it's the area on the surface of solid okay we live on the earth right we live on the earth which is an idealized sphere or better yet we can kind of idealize the earth as sphere obviously the earth has huge huge volume but the surface of the earth it self has area to it alright quite bit of area we're not going to get into that right now let's find the area of each one of these objects now I'm gonna number them that's really good way to kind of keep track of surface area because often with surface area there are lot of calculations you have to do you want to make sure you don't leave out one of the faces right but here we go right so for the first one right I'm gonna put an with little one down here that's triangle and we know I'll know that it's area is gonna be one half times its base times its height all right right we could do one half times eight and get four and four times six is gonna be 24 square centimeters now one of the great things is that at least with surface area lot of times areas repeat themselves because you definitely have these two bases that are identical so area number two will also be one-half times eight times six and that'll also be 24 square centimeters now for the other ones well all the other ones are rectangles so their areas are quite easy the area of three is going to be two times ten which is going to be 20 square centimeters the area of the third one is going to be two times six or twelve square centimeters and the area of the fourth one is going to be two times eight or sixteen square centimeters all right we have literally found five areas for the five phases that enclose our right triangular prism now all we need to do is add them all up right and that is always annoying you know there's it's it's never particularly fun but could kind of put them over here 24 24 2012 16 let's see if can add these all up without making the mistake 6 plus 2 is 8 plus 4 is 12 plus another 4 is 16 1 plus 2 is 3 plus another 2 is 5 plus another 2 is 7 8 9 looks like have 96 and specifically have 96 square centimeters that's it right surface area in theory is an easy idea right every single face that encloses prism or any figure that we're working with right is two-dimensional and they're always going to be rectangles parallelograms triangles things that you can find the area of especially after the last unit once we find the area of all the faces we simply add them all up and that's our surface area now lot of times these calculations become become annoying long adding them up can take space don't hesitate to have spare piece of paper at your side where you can be doing these calculations with lot of room because if our worksheet doesn't give you enough room for goodness sakes get out spare piece of scrap paper and do the work there all right let's do little bit more work with surface area first let's formally define it what is surface area surface area is the sum of all the areas of the polygons that make up the faces of solid that's it so whatever the faces are you can simply add up all their areas and you've got the surface area now useful tool to help visualize surface area of solid is what's known as net okay and we can think of surface area net as two-dimensional shape that comes from unwrapping solid so let's let's take look at this idea of net in exercise number two let's consider the simple cube shown how many faces does cube have each face is what type of figure all right this is easy and we talked about it in the last lesson all right rectangular prisms boxes all boxes have six faces right and you know you can count them one on top one on bottom one on the left one on the right one on the front one on the back right there's six of them so we've got six of them and they're all in the case of cube they're all squares all right every single one of them is an identical square right on 3x3 square now letter draw net of the cube there are many imagine making cuts along certain edges that would allow you to unfold the faces without them falling apart all right so like literally write net basically is like taking pieces of paper right kind of gluing them to each side and then letting them unravel as if they were wrapping paper with absolutely no waste because you know with wrapping paper don't know about you with me there's lot of waste as start to wrap present right but on this literally can unfold it what all oftentimes do when think about this is all think about keeping the base that it's literally sitting on just kind of that's the first thing draw right then sort of think about unwrapping the sides so maybe let that side fall down and let that side fall down and maybe let the top fall down afterwards right and then that gives me something like this now remember at the end of the day there should be six of these squares right what I'm doing is I'm unwrapping the faces now there are two final ones right and if you will they're the front or them and the back those could literally go anywhere right they just have to kind of go wherever you want attached to here now again that is one of many different surface nets for this you could actually take these two squares you could put that one there you could put this one all the way over here you could put it over here it doesn't really matter because when you wrap this thing up those two that forms sort of the front and the back can go in variety of different locations and it's still going to fold up properly all right but if you think about just kind of like folding up all these sides eventually you'd be able to take these two fold them up and they would sit now why are surface nets helpful you know for some students they aren't alright but they are required part of the curriculum alright we didn't use one in the last problem we could have but we didn't but what they do do is they give you two-dimensional object that you can now find the area of right that will give you the surface area of this three-dimensional object so let us see what is the common area of each face what is the surface area of each cube are sorry what is the surface area of the cube not each cube there's only one cube up here so this is actually very easy surface area problem right each one of these squares measures 3 inches by 3 inches so what is the area let's say of that individual square pause the video now and write that down all right well the common area of each face all of them being squares right is going to be 3 times 3 or 9 square inches that's the common area of each face but the beautiful thing about cube mean the spectacularly nice thing about it is it's got 6 faces that are all identical so the surface area of the cube then we oftentimes abbreviate surface area with capital capital hopefully for obvious reasons will be 6 times 9 or 54 square inches and cube again cube is the nicest object to find the surface area quite frankly because you find the area of one of the six faces and then you multiply by six because all the faces are identical so if we add piece of paper that was 54 square inches in theory we could wrap this cube with it alright let's take look at one that's little more challenging pyramid we won't be doing lot of work with pyramids in terms of surface area but little bit here and there so let's take look at it exercise number three rectangular pyramid is shown with square base and four identical isosceles triangles draw net below for this solid alright this is really cool right now first before we even draw the net let's make sure that we understand how many faces this thing has right it's got square as its base that's eight centimeters by eight centimeters and know that may not exactly look like square but they did say it's got square base right eight centimeters by eight centimeters and then we've got four triangles right that are isosceles meaning they've got two sides that are the same length each so what does this thing look like if unfold it well originally might draw that square right and that's gonna be eight by eight it's good to have those kind of dimensions there could also put them on the outside of the square the only reason didn't is because then if start to unfold the pyramid get an object that looks like this right now what do know about those triangles besides the fact that their base is eight centimeters long know that their height is ten centimeters and maybe that's better shown I'm the one that up here that's vertical right won't draw that in for each one of them because they're identical okay so I've got this square base and then I've got these four identical triangular lateral faces that was mouthful so let's now take look at letter find the area of each polygon in the net and sum to find the surface area of the solid awesome I'd like you to pause the video and see if you can figure out the surface area of that solid alright well we've got the square right so let's let's do that one maybe I'll label it like this the square area is equal to 8 times 8 or 64 square centimeters now there are 4 identical triangles alright let's just find the area of one of them well that's gonna be 1/2 times its base let me just bring up here right well actually I'll draw it down here right we've got these triangles whose bases are 8 and whose heights or 10 so it's gonna be 1/2 times 8 times 10 1 times 1/2 times 8 is 4 4 times 10 is 40 40 square centimeters now keep in mind though we do have 4 of these right now you could either say well I've got 64 plus 40 plus 40 plus 40 plus 40 or you could do the 40 times 4 get their total area so maybe I'll do that total triangle area equals 4 times 40 which is 160 square centimeters so now my surface area will be hundred and 60 plus 64 right and that's going to be 224 square centimeters let me make sure everyone keep is clear about the fact that that is my surface area but again all we had to do was simply find the area of all five faces right 64 square centimeters for that square and then 40 square centimeters for each of the triangles 40 80 120 160 in total 160 plus 64 gives me total surface area of 224 square centimeters each one of the calculations is pretty easy on its own it's just making sure that we've kept track of it all and that's where net can be helpful right because it can really say to us okay I've got to remember that have five areas that adding up here and it's little bit easier to maybe look at this picture than it is at this picture alright let's wrap this up so today we learned about the idea of surface area and it should make lot of sense right area is measurement of the space inside of two-dimensional figure and although we're working with three-dimensional figures what encloses the three-dimensional shape are two-dimensional polygons right things like rectangles and triangles and squares okay and we can find their area and when we add up all the area of all the faces that enclose solid we get the surface area net right surface area net which is just an unwrapping of all the faces and kind of laying them flat so that we can really see them helps us visualize that two-dimensional surface that's really wrapping around our three-dimensional solid it can also help us in simply keeping track of all the different areas that we might have to add up in order to find surface area all right we'll work little bit more with surface area in the next lesson for now just want to thank you for joining me for another engine math 6 lesson by emathinstruction my name is Kirk Weiler and until next time keep thinking and keep solving problems