Get ready for AP® Calculus
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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- How are you supposed how to do half-spheres and rectangles when he never showed us? I cannot get the practice problems. Does anyone have any tips on how to figure it out?(14 votes)
- Cut out the shape from a piece of paper and tape it to a toothpick (or pencil). Roll the toothpick between your hands so it spins quickly. As it moves, what shape do you see?(7 votes)
- How does something 2D become 3D?(2 votes)
- By rotating it around an axis, not a point. The best way I can think about it is to talk about a magazine. While it is not 2D, when it is closed, it is flat and fairly skinny. But if I open it up and put the front and back cover together, and try to fan out the pages as much as possible, it becomes much more of a 3D object sort of in the shape of a cylinder. Thus, my rotation would be around the spine of the magazine.
Hope this helps.(16 votes)
Is rotation when you flip it or reflection?(3 votes)
- Rotation, his goal is to turn it 3 dimensions, not to reflect it. Hope this helps! 😊(4 votes)
- What solid figure can have the same cross section as a sphere(4 votes)
- A cylinder or a cone, assuming the cross-section is parallel to the base.(2 votes)
- So the summery is that it's only the 2D shape keep turn around from the middle line to get a 3D shape?(2 votes)
- Yes. Middle line, you mean axis. The axis can be the z-one, the x-one or the y-one. I guess there are more axis, but those are the ones I know.
When it's through a point, it doesn't make a 3D Shape.(1 vote)
- When you are rotating the right triangle around the line and you get a cone, wouldn't the bottom of the cone be open? So it's not even a closed three-dimensional object?(1 vote)
- Actually if you think about it the bottom of the right triangle when it rotates would create a circle so it would be closed(3 votes)
- what is axis of rotation(1 vote)
- The straight line through all fixed points of a rotating rigid body around which all other points of the body move in circles.(2 votes)
- the cross section of the cone formed would be a triangle with a base of 6 right?(1 vote)
- You would have to say the cross section of a cone perpendicular to the base and through the vertex to be a triangle. having a base of 6 does not have enough information to be used for anything, is that the diameter or the radius? Plus, it has no meaning to the cross section unless you are trying to find the area of the triangle defined above, in which case you would also have to know the height of the cone.
Other cross sections of cones would be circles (cut parallel to the base), a hyperbola (cut perpendicular to base not at the vertex) and an ellipse or parabola if you cut along a base that intersects the plane of the base at an angle (parabola if it intersects within the base of the cone and a ellipse if it intersects the plane outside of the base of the cone).(2 votes)
- What do you get if you rotate a triangle that doesn’t have a right angle? Like I know what it would look like, but is there a term for the shape? Thanks! <3(2 votes)
- i think it's just a triangle in a different arrangement.(1 vote)
- My question show me a triangle and says that its either a cone or cylinder by 1 unit of the/ Line m? is it possible that my answer is going to be a cylinder?(1 vote)
- no you cannot cross-section a cylinder to get a triangle, but a cone can be cut parallel to the base through the vertex to get a triangle.(2 votes)
- 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.