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## Geometry (all content)

### Course: Geometry (all content)>Unit 8

Lesson 7: Cross sections of 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'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.
• 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.
• how do you find the volume of a pyramid?
• Length times width times height divided by 3
• 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.
(1 vote)
• 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 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.