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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

Teddy knows that a figure has a surface area of 40 square centimeters. The net below has 5 centimeter and 2 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 5 centimeter edges. So this is one of the 5 centimeter edges right over here. And we know that it has several other 5 centimeter edges because any edge that has this double hash mark right over here is also going to be 5 centimeters. So this edge is also 5 centimeters, this is also 5 centimeters, this is also 5 centimeters, and then these two over here are also 5 centimeters. So that's 5 centimeters, and that's 5 centimeters. And then we have several 2 centimeter edges. So this one has 2 centimeters. And any other edge that has the same number of hash marks, in this case one, is also going to be 2 centimeters. So all of these other edges, pretty much all the rest of the edges, are going to be 2 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. It looks pretty clear this is going to be a rectangular prism. But let's actually draw it. So if we were to-- we're going to fold this in. We're going to fold this that way. You could view this as our base right over here. We're going to fold this in. We're going to fold that up. And then this is going to be our top. This is the top right over here. This polyhedron is going to look something like this. So you're going to have your base that has a length of 5 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 could put the double hash marks if I want. 5 centimeters, and that's of course the same as that dimension up there. Now, when we fold up this side-- we'll do this in orange, actually-- when we fold up that side, that could be this side right over here, along this 2 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 5 centimeters in this dimension. It is 2 centimeters tall, and it is 2 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, so 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 going to be 5 centimeters times 2 centimeters. So it's going to be 10 square centimeters. Same thing for this one. It's going to be 5 by 2, 5 by 2. This one is 5 by 2. So these are each 10 square centimeters, and so is this one. This is 5 long, 5 centimeters long, 2 centimeters wide. So once again, that's 10 square centimeters. Now, these two sections right over here, they're 2 centimeters by 2 centimeters. So they're each going to be 4 square centimeters. So what's the total surface area? Well, 10 plus 10 plus 10 plus 10 is 40, plus 4 plus 4 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.