2D Cross-Sections of 3D Objects
Slicing a 3D solid with a plane creates a cross-section — a 2D shape whose form depends on the angle and position of the cut.
Introduction
Past Knowledge
3D shapes (12.1.1). 2D shape properties (Units 5–9).
Today's Goal
Identify cross-sections of prisms, pyramids, cylinders, cones, and spheres.
Future Success
Cavalieri's principle (12.3.1), calculus (integration by cross-sections).
Key Concepts
Common Cross-Sections
| Solid | Horizontal Cut (⊥ to axis) | Vertical Cut (∥ to axis) | Diagonal Cut |
|---|---|---|---|
| Rectangular Prism | Rectangle | Rectangle | Parallelogram |
| Cylinder | Circle | Rectangle | Ellipse |
| Cone | Circle | Triangle | Parabola/Ellipse |
| Sphere | Circle | Circle | Circle |
| Square Pyramid | Square (smaller) | Triangle | Trapezoid |
Worked Examples
Identify the Cross-Section
A cone is cut by a plane parallel to the base, halfway up.
Horizontal cut ⊥ to axis → Circle (smaller than the base)
Circle
Cone Through the Apex
A plane cuts through the apex of a cone and through the base diameter.
Vertical cut through axis → Isosceles triangle
Triangle (isosceles)
Common Pitfalls
Assuming All Cuts are Horizontal
The cross-section shape depends on the cutting plane's angle. A diagonal cut through a cylinder gives an ellipse, not a circle.
Real-Life Applications
MRI & CT Scans
Medical imaging takes cross-sections of the body. Each slice is a 2D image of a 3D structure.
3D Printing
3D printers build objects layer by layer — each layer is a cross-section of the final object.
Practice Quiz
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