Rotating 2D shapes in 3D | Perimeter, area, and volume | Geometry | Khan Academy

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Geometry on Khan Academy: We are surrounded by space. And that space contains lots of things. And these things have shapes. In geometry we are concerned with the nature of these shapes, how we define them, and what they teach us about the world at large--from math to architecture to biology to astronomy (and everything in between). Learning geometry is about more than just taking your medicine ("It's good for you!"), it's at the core of everything that exists--including you. Having said all that, some of the specific topics we'll cover include angles, intersecting lines, right triangles, perimeter, area, volume, circles, triangles, quadrilaterals, analytic geometry, and geometric constructions. Wow. That's a lot. To summarize: it's difficult to imagine any area of math that is more widely used than geometry.

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Closed Caption:

- 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.