Solids of Revolution
Spin a 2D shape around an axis → get a 3D solid. A rectangle becomes a cylinder, a triangle becomes a cone, a semicircle becomes a sphere.
Introduction
Past Knowledge
Cross-sections (12.1.2). Rotational symmetry (4.2.1). Basic 3D shapes.
Today's Goal
Identify the 3D solid formed by revolving a 2D shape.
Future Success
Volume formulas (12.3), calculus volumes of revolution.
Key Concepts
Revolution Table
| 2D Shape | Axis | 3D Solid |
|---|---|---|
| Rectangle (edge on axis) | One edge | Cylinder |
| Right triangle (leg on axis) | One leg | Cone |
| Semicircle (diameter on axis) | Diameter | Sphere |
| Rectangle (off-axis) | Parallel line | Hollow cylinder (annulus) |
| Trapezoid (leg on axis) | One leg | Frustum (truncated cone) |
Worked Examples
Revolved Rectangle
A 3×5 rectangle is revolved around the 5-unit side. What solid forms? What are its dimensions?
Cylinder with r = 3, h = 5.
Cylinder, r = 3, h = 5
Revolved Triangle
A right triangle with legs 4 and 7 is revolved around the 7-unit leg.
Cone with r = 4, h = 7.
Cone, r = 4, h = 7
Common Pitfalls
Which Side is the Axis?
Rotating a rectangle around its SHORT side gives a different cylinder than rotating around its LONG side. The axis determines whether the short side becomes the radius or the height.
Real-Life Applications
Pottery Wheels
A potter's profile (cross-section of the pot) is revolved on the wheel to create the 3D vessel — a literal solid of revolution.
CNC Lathes
Manufacturing revolves a profile around an axis to machine parts (bolts, axles, bowls).
Practice Quiz
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