 In the video for practice problem 4, we're asked to find the approximate surface area of a composite figure. We have a pyramid, a rectangular pyramid that sits on top of a rectangular prism. And these you just want to make sure you're keeping your work kind of in order because there is a lot going on here. Before we begin though, keep in mind that this pyramid, when we set it on top of the prism, the base of the pyramid is covered up and so we're going to have to subtract out the base of the pyramid because that's no longer going to be considered surface area because it's no longer showing. Same thing for the rectangular prism. When we set that pyramid on top, this top base is going to be covered up. So we're going to subtract out one base of the rectangular prism. So I would suggest that you write these out and do them separately. We're going to find the area of the pyramid and then subtract out the base. And then we're going to add that to the area of the prism and we're going to subtract out just one of the bases because that's covered up. So I'm going to start by finding the area of the pyramid and we're going to always start with the formula. The formula for the surface area of a pyramid is one half perimeter times slant height plus the base. And remember we want to subtract out the base. So I'm just going to go ahead and cross that out. That's what the B represents is the area of the base of the pyramid. So now we're just going to find one half perimeter times the slant height. I'm going to go ahead and put those values up here because that's what I need to find and then we'll go ahead and get started. I'm going to start with the perimeter. So if I'm just working with the pyramid here, the perimeter of the base is the distance around the square base there. So we know that 10 and 10 are these values for the base of my pyramid. And then of course it's 10 all the way around. So the perimeter is just going to add those up. 10 plus 10 plus 10 plus 10. The perimeter of the base of the pyramid is 40. And now we need to find the slant height. And this is where you want to be careful. 13 does not represent the slant height of the pyramid. 13 is actually the value of the edge of one of those lateral faces. And so in fact, if I just pull out one of the lateral triangle faces, I'm just rewriting this down here to give us a little bit more room. 13 is this value right here. And remember, slant height goes from the top and comes straight down perpendicular on one of those lateral faces. And so the slant height right here is what I need to find. So I know so far that the hypotenuse of that right triangle is 13. The other thing I know is this whole side of one of the bases of the pyramid is 10. And so if I know this whole measure is 10, if I'm looking at just this right triangle here, because we always want to make right triangles, that's going to be 5. And now that I have some information there, I can go ahead and solve for the slant height L. And if I know two sides of a right triangle, I can use the Pythagorean theorem. Side squared plus side squared equals hypotenuse squared. And if you wanted to change this to an x, if that makes more sense to you, that's fine. That just represents the value we're trying to find. And when we simplify that, we get x squared plus 25 equals 169 or x squared equals 144. And if you're familiar, 144 is a perfect square. When you take the square root of it, we get x equals 12. And so 12 will be our slant height for our pyramid. And now that we have the values, we can go ahead and plug them into our formula down here. The surface area of my pyramid, we're not going to figure out the base because we're subtracting that out. I'm just going to say 1 half times the perimeter of 40 times the slant height we found of 12. And when you plug that in your calculator, you can do 0.5 times 40 times 12. And that's going to give me 240 is the surface area of my pyramid minus the base. So we're going to keep this 240 in mind. And now we're going to go and solve for the prism surface area. I'm going to do that on a separate page to give myself more room. And remember, we're finding the area of the prism, but we have to subtract out one of the bases because it's covered up by that composite figure. So I'm going to change this 2B to just 1B. That means we're just finding the area of one of the bases. And so again up here, I'm going to write the values I need. I need the perimeter of the base. The height of the prism and the area of the base. And let's see what we have here. For the prism, this is my base 10 by 10. This is going to be the same base as the prism. And so we already found out that the perimeter of that base or the same base as the pyramid. The perimeter of that base for the pyramid was 40. And that's going to be the same for the perimeter of the prism. And then I'm going to skip down here to the area of the base because the P and the B both are going to deal with the base of that prism. The area of the base and the area of that square is just going to be 10 times 10, which is 100. And then the last value I need here is the height of the prism. And that's represented by the 14, the distance between the two bases of the prism. I'm going to go ahead and plug those values down here into my formula. Perimeter times H, 40 times 14 plus 1B, not 2Bs. And so when I simplify that, 40 times 14 is 560. And adding the 100, I get 660. And that represents the area of the prism. And I subtracted out one of the bases. And so for my last step, I'm just going to put those two values together. The area of the pyramid minus my base, what I found was 240 square yards. And then I'm going to add the area of the prism minus one base that I just found, which was 660. And if I keep them kind of separate, that will help me keep organized here. And my final answer is 900 yards squared.