What Are Cross Sections?

What the cutting plane changes

Cross sections are important because one solid can produce many different flat shapes. A cylinder can yield circles or rectangles in common settings, and a cone can produce several conic-section curves under special cuts.

This page keeps the solid and the slice together so the 2D result stays tied to the 3D geometry that created it.

A horizontal slice through a cylinder is a circle, while a vertical slice through its axis is a rectangle. The same solid gives different cross sections because the cutting plane is different. Always picture the plane's direction before naming the result.

A cone sliced parallel to its base gives a smaller circle, while a slice through its apex can give a triangle. Cross sections are therefore a way to ask how a three-dimensional object looks from one carefully chosen flat viewpoint.

• A cross section is the 2D shape formed when a solid is sliced by a plane.
• A cross section is always a 2D figure, even though it comes from a 3D object.
• Changing the plane changes the resulting section, sometimes dramatically.
• Cross sections connect solid geometry to plane geometry and, in some cases, to conic sections.

Where cross sections help

• Use cross sections when identifying the flat shape produced by slicing a prism, cylinder, cone, or sphere.
• Use them in architecture, engineering, medical imaging, and manufacturing drawings.
• Use them to connect 3D solids with familiar 2D geometry facts.

Cross-section checks

• Do not name the cross section from the outside look of the solid alone; the slicing plane matters.
• Do not forget that the result is two-dimensional, not a smaller 3D solid.
• Do not assume one solid has only one possible cross section.