Surface Area of Cones and Pyramids

you know, cones and pyramids, they have similar shape and they have similar formula for finding the surface area. So, whenever possible, like to group formulas together just to make it easier to memorize. And think you'll find that easier, too. So, let's look at the formula. The surface area, of course, is the outer surface of the object, all surfaces. And you can see it's the area of the base. So, the capital means the area of the the bottom, the base. So, in this case, it's going to be square for this pyramid. although it could be triangular pyramid or any type of pyramid but be the area of the base. Here in this cone it's going to be circle. So you're going to find the area of the base using those formulas for that particular shape. And then you're going to do 1/2 the perimeter of the base times which is the slant height. Now the slant height's like the like the leaning height like this angled height. We don't want the overall height. We use that when we calculate the volume. We're using the leaning or slant height. So it's this one here or this one here. Okay. Now in the pyramid, you don't want to use this edge. This is called the lateral edge. Okay? We want actually the height of the triangle. So that would be this height right here. So what I've done with the pyramid in this first example is I've I've basically peeled it like banana, right? So peeled it back, okay? And laid it flat. They call that net. And what you have is you've got square here on the bottom and four triangles. Okay? So if you wanted to, you could just find the area of the square and the area of one of these triangles. Multiply that by four, add everything together, and you got it. But what the formula does is it does it in little bit more efficient way for you. So I'll show you how that works. So the area of the base here, let's write down the formula. It's going to be + 12 PL. The base is square. So 6 * 6 is 36 + 1/2. Now the perimeter of the base, that's the distance all the way around. That's going to be 6 * 4. You can see that there. So it's 24. And then the slant height, the leaning height in this case is five. Okay? Like the leaning the leaning tower of Pisa, right? Okay. So this is half * 24 is 12 * 5 is 60 + 36 is 96 units squared. Okay? Because it's area where it's two dimensional. We're just covering it up with little one by one squares. Okay? Okay. Now, for the cylinder, it's I'm sorry, for the cone, it's little bit different because you can go ahead and draw the circular base, but it's little bit tricky to, you know, draw the sides, okay, or the lateral area. If you try to unfold it, it doesn't fold flat like the pyramid. just kind of, you know, guessed at what that would look like. It's hard to really, you know, unfold this. Okay, so still the same formula though. It's the area of the base plus 12 Now, let's break this down. So, the base is circle. So, what's the formula of the area of circle? It's r^ squ, right? So, it's going to be pi * 5^ 2. Okay? Because the radius is 5 plus 12 the perimeter of the base. So, the perimeter of the base is just another way of saying the circumference. And what's the formula for circumference? Well, it's 2 pi or you could say pi time diameter. And you'll notice that the 1/2 and the two, they cancel each other out because half of two is one. And so what we're left with here is * * So in some books you'll see for cones 2 + Okay? But I've just written it this way so you have less formulas to memorize. Okay? Area of the base plus 12 pl. And so now we've got pi * the radius which is 5 and the slant height this angled height which is 13. Okay. So if we simplify this we get 5^2 is 25 * + 5 * 13 which is 65. If we add those together, we get 90 pi units squared.