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### Course: Get ready for AP® Calculus > Unit 7

Lesson 7: 2D vs. 3D objects# 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!

## Want to join the conversation?

- 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?(16 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?(13 votes)

- How does something 2D become 3D?(3 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.(25 votes)

- :48

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.(3 votes)

- i dont understand How something 2D becomes 3D?(2 votes)
- It is not exactly changing a 2d object into a 3d object, it is more related to rotating a circle and putting billions and billions of these circles together to form a sphere. This would be like taking a magazine (not quite 2D, but pretty flat and thin compared to the lengths and widths. However, people fold each page and glue the front cover to the back cover to create a tree or some other 3D figure.(6 votes)

- 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?(2 votes)
- 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.(4 votes)

- if its a rectangle how do i start to rotating the shape?(2 votes)
- Think of what pattern the endpoints will make (just like the leg of the triangle would create a circle when rotated, but the point would not move. Both ends would form congruent circles, thus creating a cylinder. So what would happen if you have a magazine and you opened it so that the front and back cover are touching? You would see each of the pages on top and bottom would be on a two congruent circles.(2 votes)

- How are you supposed to find out if it would form a double cone(2 votes)
- Since a triangle forms a single cone, I assume you would have to have two triangles whose leg going away from the line of rotation being congruent.(2 votes)

- DOes it have to be only those shapes?(2 votes)
- I am not sure which shapes are in the video, but you could rotate any shape, but there are only a few that will create known 3d shapes. Triangles form cones, squares and rectangles (along a side) form cylinders, and a half circle forms a sphere.(2 votes)

- Can you help me with this question pleaseee?

"A circle of radius 1 centered at (4, 0) is rotated about the y-axis."(2 votes)

## 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.