Question

Zachary made a vertical slice that was parallel to the left and right faces of a bread roll. What shape is the cross section, and what are its dimensions?


The shape of the cross section is a that is inches by inches.



Convince Me!

What are the shapes of horizontal and vertical cross sections of a rectangular prism, and how can you determine the dimensions of the cross sections?

Answer 1

I don’t have the roll’s numbers, so in words:

- The cross section from a vertical slice parallel to the left and right faces is a rectangle. Its side lengths are the roll’s height and its front‑to‑back depth (the same two edge lengths that make up each left/right face). So you can fill the blanks with “rectangle that is (height) inches by (depth) inches.”

Convince Me! (general rule and why)
- A rectangular prism has three pairwise‑parallel face orientations:
- top/bottom faces: length × width
- front/back faces: length × height
- left/right faces: width × height
- Any slice by a plane parallel to one of those faces produces a cross section congruent to that face. So:
- a horizontal slice (parallel to top and bottom) is a rectangle whose dimensions are length by width;
- a vertical slice parallel to the front/back faces is a rectangle length by height;
- a vertical slice parallel to the left/right faces is a rectangle width by height.
- Reason: a plane parallel to a face intersects the prism in a region bounded by segments parallel to the edges of that face, so the intersection has the same side lengths as the face (you can also see this by placing coordinates: the prism is [0,L]×[0,W]×[0,H]; the plane x = c gives the rectangle [0,W]×[0,H], etc.).

If you tell me the roll’s length, width (depth), and height I’ll fill in the numeric answer for the first sentence.