Ways to slice a cube using planar planes

An explanation of the different ways in which a cube can be sliced into a series of different geometric shapes (formed by up to six sides cut from the six faces of the cube) by means of planar planes. By Khan Academy.

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<table border="0" cellpadding="0" cellspacing="0"><tbody><tr><td rowspan="2" valign="top" width="140"><div style="border: 1px solid #999999; margin: 0px 10px 5px 0px;"><a href="http://www.youtube.com/watch?v=aSokFEpoJFM&amp;feature=youtube_gdata"><img alt="" src="http://i.ytimg.com/vi/aSokFEpoJFM/default.jpg" /></a></div></td>
<td valign="top" width="256"><div style="font-size: 12px; font-weight: bold;"><a href="http://www.youtube.com/watch?v=aSokFEpoJFM&amp;feature=youtube_gdata" style="font-size: 15px; font-weight: bold;">Ways to cut a cube</a>
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<div style="font-size: 12px; margin: 3px 0px;"><span>Ways to cut a cube.</span></div></td>
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What I wanna do in this video
is explore the types of 2D shapes

we can construct,
by taking planar slices of cubes.

So what am I talking about?

Let's say we wanna deconstruct a square:

How can we slice a cube with a plane
to get the intersection of this cube,

and that plane to be a square?

We'll imagine that that plane
we are to cut, just like this,

the square is maybe the most obvious one,

so it cuts the top right over there,

it cuts the side, right over here,

it cuts the side, I guess on the back
of the glass cube, where you will see it,

right over there, dotted line,

and then it cuts this, right over here.
So you can imagine a plane that did this,

and if I wanna draw the broader plane,
I can draw it like this.

Let me see if I can do a decent,
an adequate job

at drawing the actual,
as you see you'd say a part of the plane,

that is cutting this cube.
It will look - it could look something...

it could look something like this.
And I can even color in

the part of the plane that you could
actually see it the cube were opaque,

if you couldn't see through it.
But if you could see through it

you would see this dotted line,
and the plane would look like that.

So a square is
a pretty straightforward thing to get,

if you're doing a planar slice of a cube.

But what about a rectangle?
How can you get that?

And at any point, I encourage you:

Pause the video and try to think about it
on your own:

How can you get these shapes
that I'm talking about?

Well for a rectangle
you can actually cut like this:

So, if you cut this side like this,

and then cut that side like that,

and then you cut this side like that
- I think you'd see where this is going,

this side like that,

and then you cut the bottom,
right over there,

then the intersection of the plane
that you are cutting with;

so, the intersection, let's see:

This could be the plane
that I'm actually cutting with,

so the intersection of the plane
that I'm cutting with,

and my cube is going to be
a rectangle.

So it might look like this, and once again
let me shade in the stuff,

if you kind of view this,
if you imagine the plane

just like one of those huge blades
that magicians use to saw people in half,

or pretend like they are - they give us

the illusion of sawing people in half,

it might look something like this.

Ok! So you'll go like:
"Ok, that's not so hard to digest,

that if I intersect a plane with a cube,
I can get a square, I can get a rectangle.

But what about triangles?"

Well, once again pause the video
if you think you can figure it out,

triangles are not so bad.
You can cut this side over here,

this side right over here,

and this side right over here,

and then, this is it - of course
I can keep drawing the plane,

but I think you get the idea -

this would be a triangle.

There's different types of triangles
that you can construct.

You could construct
an equilateral triangle,

so as long as this cut is the same length
as this cut right over here,

is the same length as that upper length,

or the length that intersects
on this space of the cube,

that's gonna be an equilateral triangle.

If you pushed this point up more,
actually I'd do that in a different color,

you were going to have
an isosceles triangle.

If you were to bring this point
really, really close, like here,

you would approach having a right angle,
but it wouldn't be quite a right angle:

you'd still have these angles
who'd still be less than 90º,

you can approach 90º.
So you can't quite have

an exactly a right angle,
and so since you can't get to 90º,

sure enough you can't get near to 91º,
so actually you're not gonna be able to do

an obtuse triangle either.

But you can do an equilateral,
you can do an isosceles,

you can do scalene triangles.

I guess you could say you could do
the different types of acute triangles.

But now let's do
some really interesting things:

Can you get a pentagon
by slicing a cube with a plane?

And I really want you to pause the video
and think about it here,

because that's such a fun thing.
Think about it:

How can you get a pentagon
by slicing a cube with a plane?

All right, so here I go,

this is how you can get a pentagon
by slicing a cube with a plane:

Imagine slicing the top
- we'll do it a little bit different -

so imagine slicing the top,
right over there, like this,

Imagine slicing this backside,
like that,

this back side that you can see,
quite like that,

now you slice this side,
right over here, like this,

and then you slice this side
right over here, like this.

This could be,
and alike I'm gonna draw the plane

- yet maybe it won't be so obvious
if I try to draw the plane -

but you get the actual idea:
if I slice this, the right angle

(not any right angle - 90º -
but 'the right angle' - the proper angle.

Actually I shouldn't use 'right angle',
that would confuse everything.)

If I slice it in the proper angle,

that I'm doing
with the intersection of my plane

then my cube is going to be
this pentagon, right over here.

Now let's up the stakes something,
let's up the stakes even more!

What about a hexagon?

Can I slice a cube in a way,
with a 2D plane,

to get the intersection of the plane
on the cube being a hexagon?

As you could imagine,
I wouldn't have asked you that question

unless I could.
So let's see if we can do it.

So if we slice this, right over there,
if we slice this bottom piece,

right over there,
and then you slice this back side,

like that, and then you slice
the side that we can see right over there,

(This side, I could have made it
much straighter)

So hopefully you get the idea!

I can slice this cube
so that I can actually get a hexagon.

So, hopefully, this gives you
a better appreciation

for what you could actually do
with a cube,

especially if you're busy slicing it
with large planar planes

- or large planar blades, in some way -

There's actually more to a cube
that you might have imagined in the past.

When we think about it,
there's six sides to a cube,

and so it's six surfaces to a cube,

so you can cut
as many as six of the surfaces

when you intersect it with a plane,

and every time you cut
into one of those surfaces

it forms a side.
So here we're cutting into four sides,

here we're cutting into four surfaces
of four sides,

here we're cutting into three,
here we're cutting into five

- we're not cutting into the bottom
of the cube, here -

and here we're cutting into all six
of the sides, all six of the surfaces,

of the faces of this cube.