When you unfolded solids into nets, you saw that prisms and pyramids have flat faces. Now imagine slicing straight through a solid with a knife. The flat face you uncover is a cross-section.
Idea clave
A cross-section is the 2D shape where a flat plane cuts a solid. Each face the plane crosses gives the shape one side. A slice of a prism parallel to its base is a copy of the base; a slice of a pyramid parallel to its base is a smaller copy of the base.
Por qué funciona
The plane meets each face it crosses along a straight segment, and those segments join up into a polygon. A prism's side faces stand straight up, so a slice at any height matches the base exactly. A pyramid's faces lean in toward the top point, so slices shrink as they go up.
Worked examples
Ejemplo resuelto: A box, sliced parallel to the base
A box 6 in. long, 4 in. wide and 3 in. tall is sliced parallel to its base. What is the cross-section?
- The slice is a copy of the base: a 6 in. by 4 in. rectangle.
- Its area is $6 \times 4 = 24$ square inches, at any height.
Respuesta: A 6 in. by 4 in. rectangle
Ejemplo resuelto: A pyramid, sliced through the top
A square pyramid is sliced straight down through its top point, parallel to one edge of the base. What is the cross-section?
- The slice crosses the base along a segment and passes through the top point.
- So it has a flat bottom and a top corner: a triangle.
Respuesta: A triangle
Ejemplo resuelto: A cube with a corner cut off
A plane cuts the three edges that meet at one corner of a cube. What is the cross-section?
- The plane crosses three faces near that corner.
- Three faces give three sides: a triangle. (If the three cuts are the same distance from the corner, all three sides are equal.)
Respuesta: A triangle
| Solid | Slice | Cross-section |
|---|---|---|
| Rectangular prism | parallel to the base | rectangle, same as the base |
| Rectangular prism | straight down, parallel to a side | rectangle |
| Square pyramid | parallel to the base | smaller square |
| Square pyramid | straight down through the top | triangle |
| Square pyramid | straight down, parallel to a base edge, not through the top | trapezoid |