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

Lesson 7: 2D vs. 3D objects# Ways to cross-section a cube

Learn about all the different ways we can obtain cross-sections of a cube.

## Want to join the conversation?

- Am I the only one who finds this concept really confusing? Personally, I find it hard to visualize all the different cross-sections a particular 3D object has (not just in the context of a cube). Anyone have a trick to understand cross-sections better?(32 votes)
- If you have a real cube that you're willing to ruin with a marker, you can draw on it to help you understand the cross section better. If not, you can use a 3D online model program, such as Vectary. Alternatively, you can make cubes of jello and actually cut them so that you don't have to visualise it.(15 votes)

- can you get a regular pentagon?(13 votes)
- Mhm, a regular pentagon isn't possible.

Let me explain why.

If every side would be equal, you'd have to have sides which would at least have to be smaller than the hypotenuse or square's diagonal.

If the first cut would be smaller, you'd have to reach an asymmetrical point to get a pentagon which again would leave you with two distinct and unequal side lengths.

Another way to imagine this is a circle and a pentagon.

You'd have to have 5 equally distant points and this isn't achievable in a cube if you leave the center of symmetry, with a plane which only cuts quadrilaterals, namely the midpoint.(39 votes)

- Is there a methodical approach to slicing a 3D object to find if a particular 2D shape exists, or do you just have to brute force it?(13 votes)
- he says it at the end

you can slice 2D shapes with maximum as many sides as the cube has faces(7 votes)

- I didn't understand anything from this video...(9 votes)
- What does opake mean? (1:15)(4 votes)
- opaque means you can't see through it, for example, your hand is opaque because you can't see through it but glass isn't opaque because you can see through it.(6 votes)

- From time to time i come back to watch these 'cross-section a cube videos' and every time i don't understand them.! ugh!(6 votes)
- Is it possible to make a hyperbola, a circle, an ellipse, and a parabola with a cube?(4 votes)
- Not with a simple plane, which is where conic sections comes from, since all those are made by intersecting cones with planes.(3 votes)

- Too complicated?(4 votes)
- yes its too complicated.(3 votes)

- Okay this might help some people who are confused:

The places Sal sliced on the cube that resulted in a Pentagon and a Hexagon are different!

In the pentagon, he sliced it from the ONE OF SIDES of the cube! (Any face/side that IS NOT THE TOP FACE/BOT FACE)

In the hexagon, he sliced it from the TOP SIDE/FACE of the cube!

(Also, if you watch until the end of the video, Sal DOES say we're not cutting through the bottom side in the pentagon one. So It's REALLY IMPORTANT to watch these videos until the end, because you might miss a crucial piece of information that's necessary to piece everything together!)(5 votes) - For some reason, I thought about the Tezarack from the villain Loki, Thor's brother. :/

Has anyone thought of that?

Sal is trying to find the infinity stone inside! :o

:) :D(4 votes)- HIS DRAWINGS ARE SO NICE!!

Yeah it does look a bit like the Tesseract (the Cube)!! :)(2 votes)

## Video transcript

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.