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

We're asked what is the lateral surface area of the rectangular prism and then what is the total surface area of the rectangular prism. Pause this video, have a go at this before we do this together.

All right, now let's first focus on lateral surface area. What are they referring to? Well, when you're talking about the lateral of anything, you're talking about the sides of it. So what we're really thinking about is the surface area of this rectangular prism without considering the top and the bottom. Then when they say the total surface area, then we're going to add the top and the bottom bit to that.

Now, there are a couple of ways that we could approach it. My brain actually likes to just go side by side and say, okay, this side right over here, its surface area is 9 * 5 or 45 square cm. That's going to be the same as that one over there; that's also 45. Then I could do 8 times 5 for the back side and then 8 times 5 again for this front side.

Then add them all up together. We do that in other videos. What I'll do in this video is think about another approach to it. Think about it; if there were no top and bottom and if we were to cut this right over here and we were to open it up. Then this rectangular prism would look something like this. It would look something like this. Let me do my best at trying to draw this.

It would be in four sections. It would look like it would look something like this. Where if we were to open it up, this side would be the 9 cm right over here, 9 cm. This would be this 8 cm. Let me do a different color. This 8 cm right over here would be this 8 cm right over here.

This would be another 9 cm right over there, 9 cm. And then this right over here would be another 8 cm. This right over here, 8 cm. And that place where we opened it up, I know that orange and the red look similar. Actually, I'm trying to do the orange.

Actually, those ends would otherwise be connected. So if you're trying to figure out the lateral surface area, it's equivalent to figuring out the surface area of this. The surface area of this. What we're doing here is we can think about the perimeter of the base right over here. That's going to be the same as this length.

So the perimeter of the base is equal to the sum of all of these. And what's that going to be? 8 + 9 is 17, plus another 17 that gets you to 34 cm. And then, so to figure out the lateral surface area, which is the same thing as finding the surface area of this, we just multiply it by the height, by this 5 cm right over here, 5 cm.

You might see a formula like this: that the lateral surface area is equal to the perimeter times the height. That's all they're doing. They're just opening it up like this and saying, okay, let's find the perimeter of this base here and multiply it by the height.

So the height is 5 cm. So actually, let me just put the numbers in here. So 34 cm in for p and then 5 cm in. I'm having trouble switching colors today, 5 cm in for H. And so you get the lateral surface area is equal to 34 cm times 5 cm, and it's going to be square cm now, which is going to be equal to 30 times 5 is 150, plus 4 times 5 is 20, which is going to be 170 square cm.

Now the total surface area, we just have to add the surface area of the bottom right over here and the surface area of the top. And they're each going to have the same surface area; each going to be 9 * 8 or 72 square cm. So you're going to have that twice. So 2 * 72 is 144 square cm for the top and the bottom.

And of course, we want to now add that to 170 square cm for the lateral surface area. And you are going to get—you might be able to do this in your head—but we'll just go through it methodically. 4 plus 0 is 4. 4 + 7 is 11, and then we have 314 square cm.

And we are done. So we just wanted to expose you to this. This is one way to do it. My preference would just be to go side by side and just make sure I get each of them and then calculate it versus just memorizing formulas. I don't just like memorizing formulas unless I know where it really came from.