 Before we start the first video on developing the formula for surface area of prisms, we're going to go over some vocabulary that you're going to need to know for chapter 12. So you can either just listen or you can take notes on a separate piece of paper, but this is information that will come up on homework and quizzes and tests. Chapter 12 is all about surface area of solids. And so we want to start by talking about the difference between polyhedrons and not polyhedrons. Polyhedrons are any three-dimensional solid made up of flat faces. So you'll see that there's all straight edges on all of these polyhedrons, all of these solids. And anytime you have a curved area or a curved surface, spheres, cones, and cylinders would be considered not polyhedrons. Next thing we want to talk about is the difference between prisms and pyramids, and I'm sure you've learned this before, but just to review some of this. A prism is going to have two bases, and those bases are opposite and congruent to each other. They're parallel and congruent. And you can see on each of these that are prisms, you have an opposite set of triangles here, you have opposite pentagons here, and pyramids are only going to have the one base, and we'll always come up to the tip like that. If we want to name our prisms and pyramids, keep in mind that the names of these are going to have two names. We're going to talk about the name of the base in this first one. The base is a rectangle, and then the second word to name it is going to be either prism or pyramid, depending on which one it is. So you want to just take a quick look over these to make sure you would be able to identify these. This base, again, is a pentagon, so we would call this a pentagonal prism. These two bases are triangles, so we would call this a triangular prism, versus this one over here that also has a triangular base, but because it only has one base, we would call this a triangular pyramid, and because this only has one pentagon, we would call this a pentagonal pyramid. The last thing we want to do before we get into the notes on your note sheet is just talk about some of the vocab that's going to come up in this chapter and make sure that you have a good idea of what we're talking about when we refer to these vocab words. Bases, again, are going to be the figures that in terms of a prism, the bases, we're going to have two bases, and they're going to be parallel to each other, and they're going to be congruent faces, so we're always going to have those two congruent polygons in a prism, and then just one base in a pyramid. You've probably learned before about edges and vertices, let's just review that though. The edges are the straight lines that connect all of your solid together. In this case, we could count the edges on this rectangular prism, and we would count that there's eight lines that combined will make that rectangular prism, and if we were to count the edges of this pentagonal pyramid, we know that there's going to be five edges going up to the tip, and then five edges around on the base, so each of these would have, or this would have ten edges, and this one would have twelve edges. When we talk about vertices, vertices are the points that the edges connect to. Think of those as the pointy pieces. On this pentagonal pyramid, we would have six vertices, because you have to count this top one, and on the rectangular prism, you would count the four vertices on top and the four vertices on the bottom, and we would have eight. Lastly, a term maybe you're not as familiar with is lateral faces. A lateral face is anything that's not a base. So in this instant for the pentagonal pyramid, this pentagon is the base of the pyramid, and then you would have these five triangles around as your lateral faces. You'll hear these terms come up again and again, so make sure you're familiar with them. So let's go ahead and get started on the actual note sheet. We're going to go through some steps to show you how the formula for the surface area of a prism is derived. So to start, we're asked to draw a net of this triangular prism. And remember, a net is just, if you think of unfolding the solid and flattening it out, that's what it would look like for a net. And then we're asked to label the bases and the lateral faces. So before we do that, let's just look at our triangular prism and note a couple of things. We know that the bases, because it is a prism, are going to be the opposite congruent polygons. In this case, the triangles are the bases of this prism, and that would be clear to us because this is a triangular prism. And then the lateral faces are any of the polygons that are not bases. In this case, we have three rectangles that are going to be our lateral faces here. And so when we unfold this, we're going to have the two triangles as our bases, and then these three rectangle lateral faces that are all flattened out. Before we do that though, we are going to have to do a little bit of math to find a missing measurement. We know twelve, seven, and five, but when we unfold this, we are going to have to figure out what the length of this edge is. And you'll see why when we unfold it into the net. And luckily, this is a right triangle, so we can use the Pythagorean theorem, set up the equation, and solve for that missing side of the triangle. So when we do that, if we label that x, all we have to do to find that missing side is do the leg squared plus leg squared equals the hypotenuse squared. And we're going to get x squared equals 169, and if you notice that that's a perfect square, that gives us a value of 13. So we know the length of this edge, which is the hypotenuse of my base, is 13. So now we're going to actually go ahead and flatten out and draw the net on our grid on our note sheet. I'm going to show you what it looks like, but don't draw it yet, because I'm going to walk you through it. It's important to have the measures labeled correctly. This is what the net of the right triangular prism is going to look like, and let's just talk about a few things before you start drawing it out. Notice that it is a prism, so we have the two triangle bases that are going to be opposite each other, and then when we unfold it, we get the three rectangle pieces that make up the lateral face of that prism. And so before you just draw a random triangle and three rectangles on here, we have to make sure that the measures match up to what's given to us on this prism. A good place to start when you're drawing the nets is to draw one of the bases. We know that this triangle is 5 by 12 by 13. So you can start and try to start far enough up to give yourself lots of room and draw that short leg of 5 and then the long leg of 12. And then we're just going to draw the hypotenuse there. We can't count that because it isn't a vertical or horizontal line, but we already know that that's a distance of 13. So we can go ahead and draw that first base. Before you draw the second base down below, we want to make sure that we're giving us enough room for the three lateral face pieces of the rectangles. So I'm going to draw these three rectangles one at a time. If I want to draw this green rectangle, we know that that's going from the side of 12 and that height then is going to be 7. So there's my 12 here and I'm going to count out 7 and that's represented by the green rectangle. We want to go down 7 there. At this point, you can go ahead and add that other base because now we know how far down it is. This 7 represents the distance between the two bases and then we can add on the last two rectangles. And when we do that, let's just make sure we're measuring them correctly. I know one of the rectangles has a width of 5 and then that same length of 7. So there's my 7 and then I'm going to go out a distance of 5 here and draw that rectangle. And the last rectangle then is this big one on top and I know that's a measure of 7 by the 13 that we found. So we know this is 7 here and I'm going to count out 13 and draw that last rectangle. There's a couple things you can check to make sure that you're doing this right. If I were to fold this net back up, notice that this measure is 5 and this measure is 5. These need to be the same distance because they would fold back up to make this edge right here. Same with this 13 and this 13. Those would fold back up to make an edge and you can notice that that's the same thing all the way around that netting. The last thing we're asked to do then is to label the base or bases and lateral faces. So the bases, we said it's a triangular prism. So the two triangles are the bases and this whole big rectangle now we would consider a lateral face. So you can just do that or you can label those each individually. And now we're going to use this information to answer the next few questions and we have everything we need. The first question asks us what shape is formed with the combined lateral faces and that's what we just said all three of these rectangles form the one giant rectangle for a lateral face. Identify the height of the prism on the three-dimensional drawing and on the netting and that's where if you had labeled those as 7, that corresponds to the height of that prism. The height is the distance between the two bases. So my height equals 7 and then I'm asked what is the perimeter of one base and that's why it was important for us to label this as 5 and 12 and we use the Pythagorean theorem to find that hypotenuse piece of 13 because the perimeter of the base is just going to be the distance around one of those triangles. We're just going to add up all of those sides 5 plus 12 plus 13 equals 30 and remember when it's perimeter it's just a distance so we would just label that as 30 units. Moving on to the next few questions what are the dimensions of the lateral area and so we have all of that information here. This huge rectangle is the lateral area we know the height is 7 and then the dimension of the whole thing we're just going to add up each of these length pieces so we know 5 plus 12 plus 13 well we just added that up before for the triangle we know that that total length of the rectangle is 30 and so the dimensions of the lateral area are just 7 by 30 that's what we usually do when we talk about dimensions 7 by 30 units and so if we wanted to calculate the actual lateral area that's this whole rectangle all we're going to do is the lateral area equals 7 times 30 7 times 30 is 210 and when we're talking area we're going to do units squared and then that will bring us into finding out what is the formula for the lateral area in terms of the parts of the prism so all we're doing here is going piece by piece to drive a surface area for any prism and part of that is talking about the lateral area we just found out that the lateral area was 210 and when we did that we took this 7 times the 30 this 7 represents the height of the prism so we know one piece of the lateral area is going to be the height this 30 right here actually represents the perimeter of one of my bases because we know that 5 matches with 5 13 matches with 13 and this side is 12 and so if we folded this all up this distance of the lateral area is always going to be the same as the perimeter of my base and so I know the formula for the lateral area is going to be the perimeter of the base times the height of the prism and I'm just going to put off to the side here what each of these is la is lateral area and I believe you have this at the bottom of your note sheet p is the perimeter of the base and h is the height of the prism make sure you're familiar with those terms on the bottom of your note sheet the last piece of this then is to calculate the total surface area and to do that the total surface area we just found out that the lateral area was the perimeter times the height that was 210 units squared and now we're just going to add the two bases and so to find the area of one base our base is a triangle and we know that the formula for the area of the triangle is one-half base times height and so we're just going to plug that in remember that base times height of a triangle is going to be the two pieces that are perpendicular to each other it doesn't matter which way that triangle is facing so we know five and twelve form a right angle so that's going to be my base and height that that I use for this so one-half of five times twelve equals thirty units squared so each of these triangles the area is thirty units squared and because there's two of them when we add all of these up for the total surface area of this netting of this prism we're going to do two times the area of my base plus the two hundred and ten which which represented the lateral area the perimeter times the height and when I add that all up the total surface area of this two hundred and seventy units squared that was a lot of work to find the surface area of this prism and luckily we will not have to do that every time because all this did was give us a shortcut formula that we can plug values in next time we want to find the surface area of a prism and this is what is what you want to write down in your notes and on any formula sheet you have or if you're using the formula flip book this is going to be the formula for the surface area of any prism you want to make sure you're understanding what each of these values is it's going to be two times b b represents the area of one of the bases the area of the triangle in this case plus perimeter times the height the perimeter is the perimeter of one of the bases and then that h represents the height of the prism or the distance between the bases keep in mind these two values b and p are all about your base the area of the base and the perimeter of the base so going forward with the rest of our notes we can just shorten it up by starting with this formula and plugging in the values