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Surface area using a net: rectangular prism

A polyhedron is a three-dimensional shape that has flat surfaces and straight edges.  Learn whether or not a certain net could be folded up into a certain rectangular prism. Created by Sal Khan.

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Video transcript

- [Instructor] Teddy knows that a figure has a surface area of 40 square centimeters. The net below has five-centimeter and two-centimeter edges. Could the net below represent the figure? So let's just make sure we understand what this here represents. So it tells us that it has five-centimeter edges. So this is one of the five-centimeter edges right over here. And we know that it has several other five-centimeter edges because any edge that has this double hash mark right over here is also going to be five centimeters. So this edge is also five centimeters. This is also five centimeters. This is also five centimeters. And then these two over here are also five centimeters. So that's five centimeters and that's five centimeters. And then we have several two-centimeter edges. So this one was two centimeters. And any other edge that has the same number of hash marks, in this case, one, is also going to be two centimeters. So all of these other edges, pretty much all the rest of the edges are going to be two centimeters. Now, they don't ask us to do this in the problem, but it's always fun to start with a net like this and try to visualize the polyhedron that it actually represents, and it looks pretty clear that this is going to be a rectangular prism, but let's actually draw it. So if we were to... We're gonna fold this in, we're gonna fold this that way. You could view this as our base, right over here. We're gonna fold this in. We're gonna fold that up. And then this is going to be our top. This is the top right over here. This polyhedron is gonna look something like this. So you're gonna have your base, you're gonna have your base that has a length of five centimeters. So this is our base. Let me do that in a new color. So this is our base right over here. I'll do it in the same color. So that's our base. This dimension right over here, I can put the double hash marks if I want, five centimeters, and that's of course the same as that dimension up there. Now when we fold up, when we fold up this side... Let me do this in orange, actually. When we fold up that side, that could be this side right over here, this side right over here along this two-centimeter edge. So that's that side right over here. When you fold this side in, right over here, that could be that. That's that side right over there. And then when of course we fold this side in, that's the same color. Let me do a different color. When we fold this side in, that's the side that's kind of facing us a little bit. So that's that right over there. That's that right over there. Color that in a little bit better. And then we can fold this side in. And that would be that side. And then of course we have the top that's connected right over here. So the top would go, this would be the top, and then the top would of course go on top of our rectangular prism. So that's the figure that we're talking about. It's five centimeters in this dimension. It is two centimeters, two centimeters tall, and it is two centimeters, two centimeters wide. But let's go back to the original question. Is this thing's surface area 40 square centimeters? Well, the good thing about this net here is it's laid out all of the surfaces for us, and we just have to figure out the surface area of each of these sections and then add them together, the surface area of each of these surfaces. So what is the surface area of this one here? Well, it's gonna be five centimeters times two centimeters. So it's gonna be 10 square centimeters. Same thing for this one. It's gonna be five by two, five by two. This one is five by two. So these are each 10 square centimeters, and so is this one. This is five long, five centimeters long, two centimeters wide. So once again, that's 10 square centimeters. Now, these two sections right over here, they're two centimeters by two centimeters, so they're each going to be four square centimeters. So what's the total surface area? Well, 10 plus 10 plus 10 plus 10 is 40 plus four plus four gets us to 48 square centimeters or centimeters squared. So could the net below represent the figure that has a surface area of 40 square centimeters? No, this represents a figure that has a surface area of 48 square centimeters.