 Section 12.3 is all about cylinders and in this first video we're asked to develop the formula for the surface area of a cylinder. Keep in mind that cylinders are similar to the prisms that we just learned about because prisms had two bases. Cylinders also have two bases but this time the bases are circles so there's going to be some similarities and there's going to be some differences. Before we draw the net let's just talk about what we think it's going to look like. Think of the cylinder if you think of it as just being a soup can that's a good analogy to use. If you flattened out the soup can that is we know we're going to have two circles in our net because those are the shapes of the bases. But think of taking the label off of a soup can and flattening that out and that's going to represent this lateral face here. See if you can visualize what that would look like and then we'll walk through how we draw that out on our netting. Before you draw this out I'm going to walk through this with you to make sure that you get the dimensions correct. We know that we're going to like I said have two circles that are going to represent the bases of our cylinders. But we have to be careful and not just draw two circles and then a rectangle and be done with this. We have to have the proper dimensions. So I'd like you to just listen and walk through this with me before you start drawing this. We're going to start with one of the bases and we know that that circle has a radius of three. So when you start drawing that first base I want you to just pick a point for the center of your circle. And then a radius of three you need to go up three down three over three and over three. And you're going to have these four points coming out from the center of your circle. And that's what we're looking for for you to connect to make sure that those are drawn correctly. So choose the center choose each of these four points and then you can connect the dots to make your first base. We don't want to go ahead and draw that second base right away because we don't know how far down it is. So we're going to go next to the lateral face of our cylinder. By the way this should say not right triangular prism but that should say the net of the cylinder. So we need to figure out what the dimensions are of this rectangle that represents our lateral face. And to do that we know that the height of our cylinder is five. So this distance on our lateral face will be five but we have to figure out what the length of that lateral face is. And if you think about it in terms of the soup can if you took the label off of the soup can and flattened it out. The length of that label would be represented by that distance all the way around the base when you flatten that out. And of course we know that the distance around the base the distance around the circle is represented by the circumference of a circle. So this length of my lateral face is going to be a measure of the circumference of my circle. So let's calculate the circumference. We know that the formula for the circumference of a circle is 2 times pi times r or pi times d whatever you prefer. With my radius being 3 that's going to be 2 times pi times 3 or 6 pi. We're going to want to convert that to a decimal because we don't know what how far out to go on our length if our distance is in pi. So if you plug that into your calculator 6 times pi you get approximately 18.8. And we're not asking to be real specific on the .8 part when you count out those units on the length of your rectangle. So you can just round that up to 19 but be sure to either write 6 pi or 18.8 on top. So we know that you have that correct and we know that this distance then is 5. And the radius of the circle is 3 that we're going to label all those pieces. We're asked also to label the bases B and the lateral faces LF. We know that this is the lateral face. I'm going to put an LF there and I have two bases B and B. And we need all that information when you're asked to draw the net of a figure. We need the lateral faces and bases labeled. And we need all measurements labeled with numbers as well. Now that we have everything identified it will be easy for us to go ahead and answer all these questions to derive the formula for the surface area of a cylinder. The first question we know what shape is formed with the combined lateral faces. Remember this is the lateral face and of course that shape is a rectangle. I'm going to just label these like I had on the previous page so we can answer our questions correctly. 6 pi for the length of that lateral face and a radius of 3. Identify the height of the cylinder on the three-dimensional drawing and on the netting. On the three-dimensional drawing we see the height is 5 and on my netting that height is labeled as 5 as well. H equals 5. And what is the perimeter of one base? The perimeter again is given by the circumference of the circle. 2 times pi times r we figured out was 6 pi or 18.8. Moving on to the next few questions. And I'll label these again. You can go ahead and answer those questions as I label these. Distance is 3. Part E says what are the dimensions of the lateral area? And the lateral area is 5. And we can say by 6 pi that just means a length and width of 5 and 6 pi. And then I'm asked to calculate the exact lateral area. And when it says exact that just means leave it in pi form. So 5 times 6 pi we're going to leave it in pi form. The exact lateral area then is 30 pi units squared because it is area. And finally part G what is the formula for the lateral area in terms of the parts of the prism? And that's where we're just asked to write it as the lateral area equals the length of the lateral face. Which is the circumference 2 pi r times the width of the lateral face which is the height. This lateral area is just represented by the area of that rectangle. And that's going to be 2 pi r times the height. And lastly we're asked to calculate the exact surface area of the prism. And so we're going to put this all together and find the surface area of the lateral face. And then add the bases. And when I do that I just want you to pause for a minute and walk through this with me. This first formula here is the surface area for a prism. Remember we learned that last section the surface area of the prism is 2 times the base plus the perimeter times the height. The surface area of a cylinder is derived from this prism surface area in a couple of ways. We know that both a prism and a cylinder have two bases. And they have that vertical height, that perpendicular height between the bases. So they're going to be the same formula. But the difference is that B is represented by the area of one of the bases. And the area of a base for a cylinder is a circle. And so we're going to change this B to the area of a circle, pi r squared. We have two bases. So this represents the area of both of those bases. Two times the area of the circle. And then for the area of the lateral face of my cylinder, that's what we previously did. It's similar to the perimeter times the height of a prism. And we're going to just change that because the perimeter represents the distance around my base. And we know that that's going to be represented by the circumference of the base. The circumference of the circle is 2 pi r. So really all we did was use the formula for the surface area of a prism and tweak it so it would work for the circle bases of a cylinder. So when you're using the surface area of a cylinder, you know that you're going to have pi in the formula because that's going to be represented by those circle bases. You might want to jot down that the B and the P and the H represent the area of the circle base, the perimeter of the circle base, and the height is the distance between the bases. So if you have a note sheet that you're writing down all the formulas that we're learning in this chapter, or if you're using your formula flipbook, the formula for the surface area of a cylinder, you're going to write this down. 2 times pi r squared plus 2 times pi rh. And the only values that we need to figure out the surface area of a cylinder are going to be the radius and the height of the cylinder.