Surface Area of an Octagonal Prism

Now, what do we do with that? Okay, the first thing we're going to do is we are going to find the base area, and we're going to double it. So let's slide down here and see how you do that. We have an octagon. So I'm just looking straight down on this. And I have got to draw a triangle in here because I need to find this, right? That's my apothem from the center to the side. And when you draw that in, you get a right triangle. And because this is regular, it's actually going to bisect a central angle right here at the middle. So what I've got to do is I've got to take while 360°, and I have to divide it by 8 to get that central angle. That means this angle right here is 45°. But when I draw that apotheon, it cuts in half, so the angle is 22.5°. That's the angle we're going to be working with right here in our triangle. So let's go over here and draw in 22.5°, and we've got to get the base of this triangle, and we've got to use trig to figure out our apothem. Okay, what do we know? Well, we know the side of the hexagonal prism is 8. So that means this side right here is 8, but I've cut it in half. 

So I'm going to use four for the base of my triangle. Now, I need to know this side because that's my apothem. How do I do that? I have to use trig. So let's see. I've got my angle is here. I'll call that angle. Let's say X and this is my opposite or let's call that Y because this is my unknown. Apothem. Opposite and adjacent. Opposite and adjacent. So that means my opposite is four and my adjacent is unknown my hypothesis is my unknown. So for my angle Y I'm going to use tangent because remember tangent is opposite over adjacent. So the tangent of 22 .5° is equal to four over my adjacent or my apothem and they both happen to be a's so let's call that A-okay let's figure this out by taking the tangent of 22.5 and that gives me .41 four two all over one is equal to four over a and then I do a cross product here. So I get four is equal to .4142 times a and then I'm going to divide by .41 four two divide by .4142. So use my handy dandy calculator and I'm going to take four divided by .4142 and I get 9.657 so let's say 9.7. Let's round that off to our closest tenth. 

The apothem is 9.7. Okay, now let's go back and draw that in over here. So that is 9.7. Okay, how's that going to help us? Let's go up here and look while we have to do one half AP where a is the path and P is a perimeter. So one half the apothem, which is 9.7 times the perimeter. Okay, what's the perimeter of this base? Well, one side is 8. I have 8 sides, so 8 times 8 sides gives me 64, so that's my perimeter. And then I'm going to double that. What's 64? And then once I multiply all that out, remember I have two bases, so I got to do times two, or you can just not multiply by half to start with if you just want to cancel that half out. Okay? So let's clear that. I'm going to take my 9.7 times 64. And that means my base areas are 620.8. Okay, now I need to get my lateral area. And that's that side when I opened up that piece of paper that gave me my lateral area. So what's the formula? It's the perimeter times the height. We just figured out the perimeter. 

We had 8 sides, and they measured 8 each, so my perimeter is 64. Now what was the height of this prism? Well, the height is 12. So I'm just going to take 64 times my height of 12. And that is going to give me my lateral areas, so you can see the lateral areas the easy part, 64 times 12, and that's going to give me 7 68. For my lateral area. Now let's go back up and put it all together. So let's see, I did perimeter times the height. And that gives me 7 68. Now let's add to that. So let's add to that. Our 6 20.8, and that gives me it was hard to see that isn't it for mere angle. That gives me 1388.8 1388.8 square units because it's area, and that's how you'd find the area. Now, couple helpful hints when you're doing an octagonal prism. Your central angle is always going to be 45. And when you slice it in half, you're always going to use 22.5. 

If you have any other shape besides an octagon, let's say a pentagonal prism. Let me just flip this over and talk about that one. The base area, you're going to do the same thing. You're going to draw from the center to the vertex, and then draw the apotheon, but this time your little angle right here is going to be different. You're going to take 360 divided by 5 to get the central angle. So let's see. I'm going to take 360 divided by 5, and that gives you 72. And that's the whole central angle, right? 72. But then when you got to split it in half, so divided by two, that means that central angle you would work with would be only 36°. So the triangle that you would work with would have an angle at the top of 36, okay? And then you would have half of the side of The Pentagon. 

So in each of those different shapes, you just have to find the central angle and then half it, and then work with your trig ratio. And the trig ratio, you're going to use, given this, if you know the side, let's say in this case, the side was ten of The Pentagon. That means you use 5 for the side of the triangle. And again, you'd use your opposite over adjacent and set it up as a tangent again. Hope this video was helpful on finding surface areas