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

Lesson 7: 2D vs. 3D objects

# Slicing a rectangular pyramid

What happens when you slice vertically into a rectangular pyramid? What kind of geometric shape results? Created by Sal Khan.

## Want to join the conversation?

• At Sal talks about a vertical cut. What is a vertical cut?
• It means you're cutting up and down. A horizontal cut would be cutting from side to side.
• how would you make a 2-d decagon or a nonagon. Would you always need a 3-d decagon/nonagon or can you make it out of another shape. like if you got stumped for a while.
• If you have a rectangular pyramid with 10 sides and slide it parallel to the base, you'll get a decagon when look at what was slice. The same can be said with a decagonal prism.

Hope this helps. Good question. That might be the first time I've ever typed "decagonal prism".
• I have a few questions.

1. Do rectangular pyramid's cross section always a trapezoid?

2. When someone says that the cut is parallel to the base, is the cross section that same shape as the base is?

3. Can a rectangular prism or a triangular prism have more than one vertical cross section?
• Here are a few answers.
1. If you cut vertically down from the vertex of a pyramid, you get a triangle. You don't always have to cut a pyramid at the same angle either, straight up and down. If you cut horizontally, you would get a rectangle. If you instead cut the rectangular pyramid at an angle, you could get anywhere from a kite to a pentagon. Try thinking about what angle and position you could take a cross section at to get those shapes.
2. For a prism, the cross section that is parallel to the base will be the same size and shape as the base is. For a pyramid, the cross section will be a scaled-down version of the same shape.
3. Yes! You can vertically cut a pyramid down the center-line, where the cross section intersects the vertex. You can also cut it down vertically off to one side. For a rectangular pyramid, you would get a triangle one time and a trapezoid the other.
• I'm seriously confused here, when he cuts the pyramid into a "trapezoid" it still has the characteristics of half of a side of a pyramid. Go to and look, the trapezoid does not include the side closest to you which has the slanted angles of a pyramid side. If you look at it a certain way, then yes it's a trapezoid, but looking at it another way makes it different. Does anyone else understand or see what I'm seeing? Thanks.
• It's just your perspective on the shape. If you look at the shape from head-on (meaning you can see the cutting plane as a square), the shape looks like a trapezoid.

Extra info: If you look at the cut off shape so that the cutting plane looks like a line, you'll see a right triangle.
• This thing makes no sense providing that after you cut the figure it turned into a 2d object. Shouldn't the figure still remain 3d?
• Think of cutting and folding a 2-D (flat) piece of paper up into a 3-D box. Cutting a 3-D figure and unfolding it is the reverse of this process.

Have a blessed, wonderful day!
• At wouldn't a vertical cut be going through the vertex? Then how did he get a trapezoid at ?
• It does not have to go through the vertex, and Sal is showing it as not going through the vertex. If it went through the vertex, you would see a triangle, not a trapezoid.
• How exactly is Sal cutting the square pyramid? I don't get his description.
• He is cutting perpendicular to the base (he calls it a vertical cut), but not through the vertex of the pyramid, it is slightly off the vertex. Cutting perpendicular through vertex will have a cross section that is a triangle, and if it is perpendicular off the vertex, it is a trapezoid.
• i don't understand this topic! Help me UniKitty or anybody!!
(1 vote)
• This topic is a bit tricky to understand.
Imagine this:
A butcher is cutting something with his big wide knife. He is a bit silly and starts cutting other things. He finds a pyramid toy and cuts it from top to bottom, perpendicular to the base. When he separates the two pieces he sees that the inside shape is a triangle.
He finds another pyramid and cuts it too, but this one he cuts horizontally to the base. The inside of the halves are squares.
There are more ways to cut the shapes too, but it's a little harder to explain.
I hope this helps you!
(p.s. I made up the story myself :) )
• how do you find the volume of a pyramid?
• Length times width times height divided by 3
• how would you make a 2-d decagon or a nonagon. Would you always need a 3-d decagon/nonagon or can you make it out of another shape? like if you got stumped for a while. :/

## Video transcript

So I have a three-dimensional solid right over here. And I want to imagine what type of a shape I would get if I were to make a vertical cut. And just to refresh ourselves, or give us a sense of what a vertical cut is, imagine if this was made out of jello or something kind of fairly soft. But it's still a three-dimensional solid. And I were to make a cut-- let's say I were to make a cut right over here. So let's say I had this big, sharp metal thing-- let me draw it like that. So you have this big, sharp metal thing. Let me draw it a little bit neater. So you have this big, sharp metal thing that I'm going to cut right over here. So this is the thing that I'm going to make the cut. And I'm going to go straight down. This is a vertical cut that we're talking about. So this is the thing that I'm going to cut with. Let me make it big enough so that it can capture the shape that will result. So this thing right over here, it's right in front. And I'm going to cut-- I'm going to make it go straight down and cut through this jello, or whatever you want to call it, this rectangular pyramid of jello. And what would be the resulting shape of the intersection between the jello and this thing that I'm using to cut it? And now I encourage you to pause your video and think about what the resulting shape would be. And the shape would be in two dimensions. Because this purple surface is a two-dimensional-- you could view it as part of a plane. And so where this intersects when you cut down this rectangular pyramid is the shape we're looking for. So I encourage you to pause the video and think about it or try to come up with it on your own. So let's think about it. And let me draw the rectangular pyramid again. So that's the same one. And now let me see what it would look like once I've done my cut, once I've brought this thing down. So then this is where I cut. So I cut it right over here. And then it'll exit the bottom. It'll cut along this side like that, cut along that side like that. And then it'll exit the bottom right over there. And so let me draw my whole thing. And so once I slice it down, it will look like this. My best shot at drawing it-- it will look like this. This is a vertical cut. So I've brought this thing down. And now the intersection between the thing that I'm cutting with and this pyramid is going to be this shape right over here. It cut into the top right over there. It would get all the way to the bottom right over there. And along this side, it would cut right there. And along that side, it would cut right over there. So what's the resulting shape in two dimensions of essentially the intersection between the slicer and the jello? Well, it would be this thing. It would look like a trapezoid. Let me do that in a new color. I'm overusing that one color. It would look like a trapezoid. So this would be the resulting shape. So the resulting shape would look like this. Just like that, if you made the cut right over there.