In this lesson:
By the end of this lesson, you will be able to:
Last lesson: slicing a 3D solid reveals 2D cross-sections
Today: rotating a 2D shape builds a 3D solid
These are inverse operations — two sides of the same relationship
You already know the solids. Today you learn where they came from.
A solid of revolution is a 3D object created by rotating a 2D shape about an axis
Think of a pottery wheel: spin the clay profile, get the 3D vase.
Rotate a rectangle about one of its edges — you get a cylinder
Right triangle about one leg → Cone
Semicircle about its diameter → Sphere
A right triangle has a vertical leg of length 8 and a horizontal leg of length 3. It is rotated about the vertical leg.
What solid is produced?
A) Cylinder with height 3, radius 8 B) Cone with height 8, radius 3 C) Cone with height 3, radius 8
Sketch the rotation in your mind before choosing.
B) Cone with height 8, radius 3 ✓
The leg on the axis becomes the height; the perpendicular leg becomes the radius.
The same 2D shape rotated about different axes produces different solids
Watch out: The axis determines the shape — not just the 2D profile
A full circle rotated about an external axis produces a torus (donut shape)
Everyday example: a donut, a life ring, a tire inner tube.
A right trapezoid has a perpendicular height of 6, a base of 4, and a top of 2. It is rotated about its perpendicular side (height = 6).
A) Cylinder B) Frustum (truncated cone) C) Cone
Imagine the shape sweeping around the axis — then advance.
B) Frustum ✓
A frustum is a cone with its top cut off — produced by rotating a trapezoid.
To find the 2D profile that generates a solid of revolution:
Technique: Slice the solid in half along its axis of rotation → the cut face IS the 2D profile
A frustum has base radius 5, top radius 2, and height 7.
Sketch the 2D profile that generates it.
On paper: draw the shape with the axis marked. Then advance.
The profile for a frustum with base radius 5, top radius 2, height 7:
The perpendicular side goes on the axis — rotation is about the height
Key insight: Every horizontal cross-section of a solid of revolution is a circle (or annulus)
The two lessons complete each other.
At height above the base, a solid of revolution has a circular cross-section
Volume ≈ sum of all cross-sectional areas × (tiny thickness)
This is integration — you'll formalize it in calculus. The geometry is what you're learning now.
✓ A solid of revolution is built by rotating a 2D shape about an axis
✓ Rectangle → cylinder; right triangle → cone; semicircle → sphere; trapezoid → frustum
✓ To find the profile: slice the solid along its axis — the cut face is the 2D profile
✓ All horizontal cross-sections of a solid of revolution are circles
"The axis doesn't matter" — the same 2D shape on a different axis gives a different solid
"Revolution only gives smooth shapes" — cylinders and cones ARE solids of revolution; flat faces come from rectangular/triangular profiles
"Any rotating shape gives a cone" — only a right triangle about one leg gives a cone; rectangles give cylinders
"Cross-sections of a solid of revolution can be any shape" — horizontal cross-sections are always circles
Both directions now complete:
Coming up:
The geometric intuition you've built here will carry through all of these.
Click to begin the narrated lesson
Identify cross sections