The Tombstone Theorem Diagrams

The Tombstone Theorem

Visual Proof: How Geometry and Calculus Tell the Same Story

Diagram 1: The Archimedean Configuration

A cylinder of radius r and height r contains a hemisphere (dome at bottom) and a cone with its point DOWN at the center of the base, opening upward. This creates the key geometric relationship.

Cylinder

Hemisphere

Inverted Cone

Diagram 2: Cavalieri's Principle - The Slice at Height h

At any height h from the top, observe the cross-sectional areas. Move the slider to see different slice positions.

Slice Height (h): 0.50r

At height h:
Cylinder area = πr²
Cone area = πh²
Sphere area = π(r² - h²) = πr² - πh²

KEY INSIGHT: Sphere Slice = Cylinder Slice - Cone Slice
This is true at EVERY height. Therefore, by Cavalieri's Principle, the volumes must follow the same relationship.

Diagram 3: The 1:2:3 Volume Ratio

For a cylinder of radius r and height 2r (containing a full sphere), the volumes form a perfect integer ratio.

The Eternal Ratio:
Double Cone : Sphere : Cylinder = 1 : 2 : 3

V(cone) = ⅓πr²(2r) = ⅔πr³
V(sphere) = ⁴⁄₃πr³
V(cylinder) = πr²(2r) = 2πr³

Ratio: (⅔πr³) : (⁴⁄₃πr³) : (2πr³) = 1 : 2 : 3

Diagram 4: Integration = Stacking Slices

The integral symbol ∫ is just an instruction to add up infinitely many slices. Adjust the number of slices to see how integration works.

Number of Slices: 10

CALCULUS DECODED: When you integrate x² from 0 to r, you get x³/3. This isn't algebraic magic—it's measuring how "pointy" shapes (pyramids, cones) fill exactly ⅓ of the space that "blocky" shapes (prisms, cylinders) fill.