Relating 2-D and 3-D Geometry

A cross-section is the intersection of a plane and a solid.

For example, if one cuts a right circular cylinder with a plane parallel to one of the bases, one obtains a circular cross-section. On the other hand, if one cuts that same cylinder with a plane perpendicular to the bases, one sees a rectangular cross-section. Furthermore, if the plane is neither parallel nor perpendicular to the bases, one obtains an elliptical cross-section (or possibly a partial ellipse and one or two straight segments, if the plane cuts one or both of the bases). Notice that the cross-section is heavily dependent on the angle and position of the plane.

Cross-sections are important because they reduce the number of dimensions that one must deal with. We are generally not very comfortable dealing with problems in three or more dimensions, since the surface of the paper that we use is two-dimensional. However, our minds can more easily grasp higher dimensions when we look at a simpler figure first, and then expand our reach once we fully understand the basics.

Many 3-D geometry problems can be solved by taking the correct 2-D cross-sections, and understanding a solid's cross-sections can help us visualize the solid.

A cross-section is the area of the shape where a 3-D shape meets a 2-D plane. Think of it as slicing an object in half and looking at where it was cut.

For example, cross-sections of some certain shapes have names. A cross-section of a sphere is always a circle. A cross-section of a cylinder could be a circle, ellipse, or quadrilateral.