Lateral & total surface area of rectangular prisms

0:00- [Instructor] 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.

0:13What 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. And then when they say the total surface area, then we're going to add the top and the bottom bit to that.

0:31Now there's 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 nine times five or 45 square centimeters. That's going to be the same as that one over there. That's also 45.

0:46And then I could do eight times five for the backside and then eight times five again for this front side, and then add 'em 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.

1:07It would look something like this. Let me do my best at trying to draw this. It would look, it would be in four sections. It would look something like this where if we were to open it up, this side would be the nine centimeters, right over here, nine centimeters.

1:35This would be this eight centimeters. Lemme do a different color. This eight centimeters right over here would be this eight centimeters right over here. This would be another nine centimeters right over there, nine centimeters. And then this right over here would be another eight centimeters.

1:58This right over here, eight centimeters. 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. And so if you're trying to figure out the lateral surface area, it's equivalent to figuring out the surface area of this.

2:17The 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? Eight plus nine is 17 plus another 17.

2:42That gets you to 34 centimeters. 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 five centimeters right over here, five centimeters. And you might see a formula like this.

2:59The 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 five centimeters. So actually, let me just put the numbers in here.

3:15So 34 centimeters in four P and then five centimeters in... I'm having trouble switching colors today. Five centimeters in four h. And so you get the lateral surface area is equal to 34 centimeters times five centimeters, and it's gonna be square centimeters now, which is going to be equal to 30 times five is 150 plus four times five is 20, which is going to be 170 square centimeters.

3:49Now 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. They're each going to be nine times eight or 72 square centimeters. So you're gonna have that twice.

4:10So two times 72 is 144 square centimeters for the top and the bottom. And of course we want to now add that to 170 square centimeters 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.

4:27Four plus zero is four. Four plus seven is 11. And then we have 314 square centimeters, and we are done. So we just wanted to expose you to, this is one way to do it. My preference would just to go side by side, and just make sure I get each of them and then calculate it versus just memorizing formulas.

4:50I don't just like memorizing formulas, unless I know where it really came from.