Cross-sections of 3D figures
Cross-Sections of Solids
Cross-Sections of Solids is a topic in Three-Dimensional Measurement in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.10(A), which requires students to identify the shapes of cross-sections of three-dimensional figures.
A cross-section is the two-dimensional shape formed when a plane slices through a solid.
Theory
A cross-section is the 2D shape formed when a plane slices a solid. The shape depends on the solid and the direction of the cut:
- Sphere: always a circle.
- Cylinder: horizontal slice \(\to\) circle; vertical slice through the axis \(\to\) rectangle.
- Cone: horizontal slice \(\to\) circle (smaller toward the apex).
- Prism / cube: a slice parallel to the base has the same shape as the base.
Common cross-sections:
How to find a cross-section
- Picture the slicing plane through the solid.
- Trace where the plane meets each face or surface.
- Name the resulting 2D shape.
- Note that a different cut direction can give a different shape.
Any flat slice of a sphere is a circle (a great circle if through the center).
A horizontal slice is a circle; a vertical slice (through the axis) is a rectangle.
A circle (smaller as you go up toward the apex).
A square — the same shape as the face.
Common pitfalls
Frequently asked questions
What is a cross-section?
The 2D shape formed when a plane cuts through a 3D solid.
What is the cross-section of a sphere?
Always a circle, whatever the direction of the cut.
What are the cross-sections of a cylinder?
A circle for a horizontal slice, and a rectangle for a vertical slice through the axis.
What is the cross-section of a prism parallel to its base?
The same shape as the base.