Solids of Revolution

Lesson 2 of 2: Building 3D from 2D

In this lesson:

• Generate a 3D solid by rotating a 2D shape about an axis
• Identify the 2D profile that generates a given solid
• Connect cross-sections and solids of revolution

Learning Objectives

By the end of this lesson, you will be able to:

3. Identify the 3D solid generated by rotating a given 2D shape about a specified axis
4. Sketch the 2D profile whose rotation would produce a given solid of revolution
5. Connect cross-sectional thinking to volume computation and real-world applications

From 3D to 2D — and Back

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.

What Is a Solid of Revolution?

A solid of revolution is a 3D object created by rotating a 2D shape about an axis

• Every point on the 2D shape traces a circle as it rotates
• The collection of all those circles forms the 3D solid
• The axis is a line — one edge, or a line drawn nearby

Think of a pottery wheel: spin the clay profile, get the 3D vase.

Rectangle → Cylinder

Rotate a rectangle about one of its edges — you get a cylinder

Triangle → Cone, Semicircle → Sphere

Right triangle about one leg → Cone

• The leg on the axis is fixed → becomes the height
• The other leg sweeps the base circle → becomes the radius
• The hypotenuse sweeps the conical surface

Semicircle about its diameter → Sphere

• Every point on the semicircle traces a circle
• Together they form a complete sphere of radius

Quick Check: Which Solid?

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.

Answer: Cone with Height 8, Radius 3

B) Cone with height 8, radius 3 ✓

• Vertical leg (on the axis) → height = 8
• Horizontal leg (sweeps the base circle) → radius = 3
• Hypotenuse → slanted surface of the cone

The leg on the axis becomes the height; the perpendicular leg becomes the radius.

The Axis Makes All the Difference

The same 2D shape rotated about different axes produces different solids

• Rectangle rotated about long edge → short, wide cylinder
• Rectangle rotated about short edge → tall, narrow cylinder
• Rectangle rotated about a center line → thicker cylinder (same height, smaller radius)

Watch out: The axis determines the shape — not just the 2D profile

Extension: Full Circle → Torus

A full circle rotated about an external axis produces a torus (donut shape)

• The axis is outside the circle — at distance from center
• Each point on the circle sweeps a circle of its own
• The result: a ring-shaped surface enclosing a tube

Everyday example: a donut, a life ring, a tire inner tube.

Quick Check: Trapezoid Rotation

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).

What solid is produced?

A) Cylinder
B) Frustum (truncated cone)
C) Cone

Imagine the shape sweeping around the axis — then advance.

Answer: Frustum (Truncated Cone)

B) Frustum ✓

• Perpendicular side (axis) → height = 6
• Base of length 4 → base radius = 4
• Top of length 2 → top radius = 2
• Slanted side → tapered outer surface

A frustum is a cone with its top cut off — produced by rotating a trapezoid.

Working Backward: Find the Profile

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

Worked Example: Match the Profile

• Rectangle → Cylinder
• Right triangle → Cone
• Semicircle → Sphere
• Right trapezoid → Frustum

Quick Check: Sketch the 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.

Profile Answer: Right Trapezoid

The profile for a frustum with base radius 5, top radius 2, height 7:

• Perpendicular side (on the axis): length 7 — this is the height
• Bottom parallel side (base radius): length 5
• Top parallel side (top radius): length 2
• Slanted side: connects top-right corner to bottom-right corner

The perpendicular side goes on the axis — rotation is about the height

Cross-Sections Connect Back

Key insight: Every horizontal cross-section of a solid of revolution is a circle (or annulus)

• Why? Every point at a given height traces a circle during the rotation
• The circle's radius = the profile's width from the axis at that height
• This is why disk method works in calculus: volume = sum of circular cross-sections

The two lessons complete each other.

Preview: Calculus Disk Method

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.

Key Takeaways

✓ 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

Watch Out: Common Mistakes

"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

What's Next

Both directions now complete:

• Lesson 1: Slicing 3D solids → 2D cross-sections
• Lesson 2: Rotating 2D profiles → 3D solids of revolution

Coming up:

• Volume formulas and composite solid applications (HSG.GMD.A.3)
• Cavalieri's principle connects cross-sections to volume reasoning

The geometric intuition you've built here will carry through all of these.