Definition: A cross section is the two-dimensional shape you see when a plane slices through a three-dimensional solid.
What is a cross section?
Imagine slicing a loaf of bread. The flat face revealed by the cut is the cross section.
What shapes can appear?
It depends on the solid and the angle of the cut.
- Rectangular prism, cut parallel to a face: a rectangle
- Rectangular prism, cut at a slant: a rectangle, or other shapes such as a hexagon
- Cylinder, cut parallel to the base: a circle
- Cylinder, cut straight down: a rectangle
- Cone, cut parallel to the base: a circle
- Square pyramid, cut parallel to the base: a smaller square
How do you find the area?
Measure the cross section. A horizontal slice of a cylinder with radius 4 cm is a circle of area 3.14 x 16 = 50.24 square centimeters. A vertical slice through the center of a cylinder with radius 4 and height 10 is a rectangle 8 by 10, area 80 square centimeters.
What is a common mistake?
Confusing a cross section with the whole face of the solid. The cross section depends on where the plane cuts.
Example: A vertical cut through a cube of edge 5 gives a 5 by 5 square.