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Rotating 2D shapes in 3D

If you rotate a 2D shape about an axis, the shape will define a 3D object. Watch Sal rotating various 2D shapes and see what 3D objects he gets!

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Video transcript

- What I want to do in this video is get some practice visualizing what happens if we were to try to rotate two dimensional shapes in three dimensions. Well what do I mean by that? Let's say I started with a right triangle. So let's say my right triangle looks like this. So let's say it looks like that. Right over there. And so this is a right angle. And let's say that this width right over here is three units and let's say that this length is five units and now I'm gonna do something interesting. I'm gonna take this two dimensional right triangle and I'm gonna try to rotate it in three dimensions around this line, around the line that I'm doing as a dotted magenta line. So I'm gonna rotate it around this line right over there. So if I were to rotate it around this line, what type of a shape am I going to get? And I encourage you -- It's going to be a three dimensional shape. I encourage you to think about it, maybe take out a piece of paper, draw it, or just try to imagine it in your head. Well to think about it in three dimensions, what I'm going to do is try to look at this thing in three dimensions. So let me draw this same line but I'm gonna draw it at an angle so we can visualize the whole thing in three dimensions. So imagine if this was sitting on the ground. So that's our magenta line, and then I can draw my triangle. So my triangle would look something like this. So it would look like this. So once again this is five units, this is three units, this a right triangle. I'm gonna rotate it around the line, so what's it gonna look like? Well this and this right over here is gonna rotate around and it's gonna form a circle with a radius of three, right? So it's gonna form, so it intersects, if that was on the ground it's gonna be three again. And let me draw it down so it's gonna keep going down. Whoops. We don't want to press the wrong button. So it's gonna look something like this. That's what the base is gonna look like. But then this end right over here is just gonna stay at a point because this is right on that magenta line. So it's gonna stay at a point. And so if you were to look at the intersect so it would look something like this. So it would look like this and then you'd have another thing that goes like this and so if you were to take a section like this it would have a little smaller circle here based on what this distance is. So what is the shape, what is the shape that I am drawing? Well what you see, what it is, it's a cone. It's a cone and if I shade it in you might see the cone a little bit better. So let me shade it in so you see the cone. So what you end up getting is a cone where it's base, so I'm shading it in so that hopefully helps a little bit, so what you end up getting is a cone where the base has a radius of three units. So let me draw this. This right over here is the radius of the base and it is three units. I could also draw it like this. So the cone is gonna look like this. And this is the tip of the cone and it's gonna look just like this. And once again let me shade it a little bit so that you can appreciate that this is a three dimensional shape. So draw the cone so you can shade it and we can even construct the original so that, well or we can construct the original shape so you see how it constructs so it makes this, the line, that magenta line, is gonna do this type of thing. It's gonna go through the center of the base, it's gonna go through the center of the base just like that. And our original shape, our original right triangle, if you just took a cross section of it that included that line you would have your original shape. Let me do this in orange. So the original shape is right over there. So what do you get? You get a cone where the radius of the base is three units. Interesting.