Lateral & total surface area of triangular prisms | Grade 8 (TX) | Khan Academy

We're asked what is the lateral surface area of the triangular prism and what is the total surface area of the triangular prism. Pause this video and try to solve this on your own before we work through this together.

All right, so first let's just remind ourselves what lateral even means. In everyday life, lateral means to the side or if we're dealing with the sides of something. Now, when we're thinking about the lateral surface area of this rectangular prism, the way it's oriented right now, it might not be exactly obvious what they mean by the side—the lateral surface area.

The general principle when I'm thinking about lateral surface area, in particular, I like to think about: can I take this figure and can I stand it up on a base? If you did, if you took this one and you set it up so that this is the base right over here, well then it might be a little bit more obvious. It will look something like this, where this is the base and then you have a height that looks something like this, and this is the top, which has the same surface area as the base.

I know I could have let me draw it better than that. The best way is to just draw the top too and then connect these, so it will look something like this. If we make this the base where this seven is now the height, this three is now the, I guess one way we could think about it is the height of the base—that's three right over here. This length right over here is eight, and this length right over here is five.

Now, when we're thinking about surface area—and to be honest my brain likes to just go side by side and figure out all of the surface areas and add them up— and that's probably how I would have approached it if they weren't asking for the lateral surface area and if I didn't also know that the standard we're trying to cover here deals with formulas.

You might sometimes see this formula right over here: that lateral surface area is equal to the perimeter of the base times height. Now I'm not a big fan of formulas like this because you might just memorize them and then forget them the next day. But it is useful to know where this comes from, and then we can actually apply it.

So one way to think about it: the lateral surface area is the surface area of the three rectangles that essentially connect the top here, this top triangle, and this bottom triangle—or if we're looking at this orientation, it's this side, this side, and the bottom right over here.

Now when they talk about the perimeter, they're talking about this length that I am showing in red, essentially the perimeter of the base, and that base is a triangle. Then, you multiply it by the height and that makes sense because when you have the perimeter, you have this length plus this length plus that length. If you wanted to find the area of each of those panels, so to speak, you would multiply the length of that side times its height.

The length of this side times the height, the length of that side times the height. Here, we're just adding them all together first and then multiplying by the height. But let's just do that: what is the perimeter of this base right over here? P is going to be equal to this eight, so that is eight, and then this side over here is five, and then this side over here is also five—they tell us that.

So it's 8 + 5 + 5, which is equal to 18. That's the perimeter. And then what's the height here? Well, we see that the height here is seven, so that is the height. So the lateral surface area is: let me just write it this way, it is equal to 18—that's the perimeter of the base—times the height, times 7.

I'm doing that in the same color, times 7, which is going to be equal to 10 * 7 is 70, 8 * 7 is 56. 70 + 56 is 126, and they don't give us the units here, but we could say square units or unit squared, however we want to think about it.

Now another way you could think about this is if we were doing it separately. If we just did each panel by themselves, you would first do maybe you do this panel and you would say, well that area is going to be 5 * 7, and then you would do this panel over here, which is also going to be 5 * 7, and then you would do this panel down here, which is essentially this one, this big panel down there, which is 8 * 7, plus 8 * 7.

This would also calculate the lateral surface area, but notice this is the same thing. If we were to just factor out a seven here, this is the same thing as 7 times (5 + 5 + 8) or the height, this seven times the perimeter of the base. So these are just equivalent things.

But now let's get to this other part: what is the total surface area of the rectangular prism? Well, for total surface area, we just have to add the area of the base and the top to the lateral surface area. Sometimes you'll see a formula like this: that the surface area is equal to the lateral surface area—which is the perimeter of the base times the height—plus two times the base because that's the area of the base plus the top.

I really sorry, this is a plus. I really don't like formulas like this because you're going to forget what they mean; it's better to just think it through common sense. But let's just do it where you have the lateral surface area, and now the surface area of each of these triangles will be base times height times 1/2.

The area of a triangle is 1/2 times base times height, so the base here is 8 and the height is 3. So area is equal to (1/2 * 8 * 3). 1/2 * 8 is 4, and 4 * 3 is equal to 12 square units. So that's one of them. We have two of them; we're going to have the other one back there.

So if you say 126 plus 2 * 12, so 24, that will get us to 150 square units as the total surface area.