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Intro to nets of polyhedra

Admittedly, "nets of polyhedra" sounds like the title of a bad sci-fi movie about man-eating, multi-headed fish. But in reality, nets of polyhedra are just 2D objects that wrap around 3D objects, like wrapping paper around presents. Created by Sal Khan.

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Video transcript

What we're going to explore in this video are polyhedra, which is just the plural of a polyhedron. And a polyhedron is a three-dimensional shape that has flat surfaces and straight edges. So, for example, a cube is a polyhedron. All the surfaces are flat, and all of the edges are straight. So this right over here is a polyhedron. Once again, polyhedra is plural. Polyhedron is when you have one of them. This is a polyhedron. A rectangular pyramid is a polyhedron. So let me draw that. I'll make this one a little bit more transparent. Let me do this in a different color just for fun. I'll make it a magenta rectangular pyramid. So once again, here I have one flat surface. And then I'm going to have four triangular flat surfaces. So this right over here, this is a rectangular pyramid. Now, it clearly looks like a pyramid. Why is it called a rectangular pyramid? Because the base right over here is a rectangle. So these are just a few examples of polyhedra. Now, what I want to think about are nets of polyhedra. And actually, let me draw and make this transparent, too, so we get full appreciation of the entire polyhedron, this entire cube. So now let's think about nets of polyhedron. So what is a net of a polyhedron? Well, one way to think about it is if you kind of viewed this as made up of cardboard, and you were to unfold it in some way so it would become flat, or another way of thinking about it is if you were to cut out some cardboard or some paper, and you wanted to fold it up into one of these figures, how would you go about doing it? And each of these polyhedra has multiple different nets that you could create so that it can be folded up into this three-dimensional figure. So let's take an example. And maybe the simplest example would be a cube like this. And I'm going to color code it. So let's say that the bottom of this cube was this green color. And so I can represent it like this. That's the bottom of the cube. It's that green color. Now, let's say that this back surface of the cube is orange. Well, I could represent it like this. And notice, I've kind of folded it out. I'm folding it out. And so if I were to flatten it out, it would look like this. It would look like that. Now, this other backside, I'll shade it in yellow. This other backside right over here, I could fold it backwards and keep it connected along this edge, fold it backwards. It would look like this. It would look like that. I think you get the general idea here. And just to be clear, this edge right over here is this edge right over there. Now I have to worry about this top part. Maybe it is in-- let me do it in a pink color. This top part of the cube is in this pink color, and it needs to be attached to one of these sides. I could attach it to this side or this side. Let's attach it over here. So let's say it's attached to that yellow side back here. So then when we fold it out, when we really unpack the thing, so we folded that yellow part back, then we're folding this part back, then it would be right over here. And then we could fold this front face right over here. We could fold that out along this edge, and it would go right over there. It would go right over there. And then we have one face of the cube left. We have this side right over here. Well, we could do, actually, several things. We could fold it out along this edge. And then we would draw the surface right over there. Or if we want to do something interesting, we could fold it out along the edge that it shares with the yellow, that backside. So we could fold it out like this. So if we folded it out like this, it would be connected to the yellow square right over here. So you see that there's many, many ways to construct a net or a net that when you fold it all back up will turn into this polyhedron, in this case, a cube. Let's do one more example. Let's do the rectangular pyramid, because all of these had rectangles. Or in particular, these had squares as our surfaces. Now, the most obvious one might be to start with your base right over here. Start with your base and then take the different sides and then just fold them straight out. So, for example, we could take this side right over here, fold it out, and it would look like that. We could take this side back here, and once again, just fold it out. And it would look like that. It should be the same size as that orange side, but I'm hand drawing it, so it's not going to be perfect. So that's that right over there. And then you could take this front side right over here, and once again, fold it out along this edge. So it would look like this. And then finally, you could take this side right over here, and once again, fold it out along this edge and it would go right there. But this isn't the only net for this rectangular pyramid. There's other options. For example, and just to explore one of them, instead of folding that green side out that way, instead we might have wanted to fold it out along this edge with the yellow side that you can't see. Actually, let's make it a little bit different. Let's fold it out along this side since we can see the edge. And let me color the edge. So this is the edge right over here on the blue triangle. So this is the edge. And when you fold the green triangle out, it would look like this. If you fold it the green triangle out, it would look like this. So hopefully this gives you an appreciation. There's multiple ways to unfold these three-dimensional figures, these polyhedra, or multiple ways if you wanted to do a cardboard cutout and then fold things back together to construct them. And these flattened versions of them, these things, these unpacking of these polyhedra, we call nets.