The Tombstone Theorem: Unifying the Sphere, Cylinder, and Cone

J. Rogers, SE Ohio

Subtitle: How Geometry and Calculus Tell the Same Story of 1, 2, and 3

Foreword

Archimedes of Syracuse could have chosen anything for his tombstone. He had invented war machines, discovered the principles of buoyancy and leverage, and made contributions that would echo through millennia. But when it came time to decide what would mark his grave for eternity, he chose a simple geometric figure: a sphere inscribed in a cylinder.

Why? Because he had discovered that the sphere occupies exactly two-thirds the volume of the cylinder that contains it—a relationship so perfect, so fundamental to the structure of space itself, that it transcended all his practical achievements. This paper shows why that discovery deserved to be carved in stone. The cone, sphere, and cylinder are not three separate formulas to memorize. They are one truth, expressed three ways, locked in an eternal 1:2:3 ratio that bridges ancient geometry and modern calculus.

He knew this wasn't just a geometric fact, but a window into how reality is constructed - that the relationship between different shapes, between growth and accumulation, was revealing universal principles he couldn't fully articulate yet but knew were there.

Introduction

For centuries, students have been forced to memorize three distinct volume formulas: the cylinder (

        $\pi r^2h$
      

), the cone (

        $\frac{1}{3}\pi r^{2}h$      

), and the sphere (

        $\frac{4}{3}\pi r^3$
      

). In classrooms, these formulas often appear as separate, unrelated facts.

However, they are not separate. They are parts of a single geometric system. When they are inscribed within one another—sharing the same radius and height—their volumes lock into a perfect 1 : 2 : 3 integer ratio.

This paper demonstrates that Calculus is not merely a tool for calculation, but a language for describing geometry. By examining the relationship between the "stack of squares" (the pyramid/cone) and the "stack of disks" (the cylinder), we can prove that the "magic" fraction

        $\frac{4}{3}$
      

is the inevitable result of subtracting a cone from a cylinder.

Part I: The Geometric Intuition (The Slice)

To see the unity of these shapes, we must stop looking at them from the outside and start looking at their cross-sections (slices).

We begin with a Cylinder of radius

        $r$
      

and height

        $r$
      

. Inside it, we place two shapes:

1. A Hemisphere (radius

           $r$
         

   ).

2. A Inverted Cone (radius

           $r$
         

   , height

           $r$
         

   ).

We apply Cavalieri’s Principle, which states: If two solids have slices of equal area at every height, they have the same total volume.

Imagine slicing all three shapes horizontally at a specific height

        $h$
      

(measured from the top).

1. The Cylinder Slice:
   The cylinder is a uniform block. No matter where you slice it, the cross-section is a circle of radius

           $r$
         .

   $$Area_{cylinder} = \pi r^2$$      

2. The Cone Slice:
   Because the cone has a height equal to its radius (a 45-degree slope), the radius of the slice at height

           $h$
         

   is simply

           $h$
         .

           $$Area_{cone} = \pi h^2$$

3. The Hemisphere Slice:
   This is where the geometry shines. By the Pythagorean theorem, the radius of a slice of a sphere at height

           $h$
         

   is

           $\sqrt{r^2 - h^2}$
         

   .

   $$Are{a}_{sphere}=\pi (\sqrt{r^2-h^2}{)}^2=\pi (r^2-h^2)$$

The Unification:
If we expand the area of the sphere slice, we get

        $\pi r^2 - \pi h^2$
      

. Notice the relationship:

        $$\text{Sphere Slice} = \text{Cylinder Slice} - \text{Cone Slice}$$

Geometrically, this proves that a Hemisphere is simply the volume of a Cylinder minus the volume of a Cone.

Part II: The Calculus (The Summation)

Calculus is simply the act of adding up these slices infinitely. It formalizes the geometric subtraction we just observed.  This is the intuition that 

Let us integrate these areas from height

        $0$
      

to

        $r$.

1. The Cylinder (The Constant):
The integral of a constant represents a block (a prism or cylinder).

        $$V_{cyl}={\int }_0^r\pi r^2\hspace{0.17em}dh=\pi r^2[h{]}_0^r=\pi r^3$$

2. The Cone (The Linear Growth):
This is the origin of the

        $\frac{1}{3}$
      

. Just as a pyramid is

        $\frac{1}{3}$
      

of a cube, the integral of a variable squared results in a division by 3.

        $$V_{cone} = \int_{0}^{r} \pi h^2 \, dh = \pi \left[ \frac{h^3}{3} \right]_{0}^{r} = \frac{1}{3} \pi r^3$$

3. The Sphere (The Difference):
We integrate the sphere's slice formula (

        $\pi r^2 - \pi h^2$
      

). Linearity of integration allows us to split this into two parts:

        $$V_{hemisphere} = \int_{0}^{r} (\pi r^2 - \pi h^2) \, dh$$              

$$V_{hemisphere} = \int_{0}^{r} \pi r^2 \, dh - \int_{0}^{r} \pi h^2 \, dh$$      

This translates directly to our geometric words from Part I:

        $$V_{hemisphere} = V_{cylinder} - V_{cone}$$      

Substituting the results:

        $$V_{hemisphere} = \pi r^3 - \frac{1}{3} \pi r^3 = \frac{2}{3} \pi r^3$$      

To get the full sphere, we double the hemisphere:

        $$V_{sphere} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3$$      

Part III: The 1-2-3 Theorem

We can now assemble the "Archimedean Box." Consider a cylinder of radius

        $r$
      

and height

        $2r$
      

(fitting the full sphere).

• The Cone: A double-cone (hourglass shape) inside this box has a volume of

          $\frac{1}{3}$
        

  of the cylinder.

• The Sphere: The sphere takes up

          $\frac{2}{3}$
        

  of the cylinder.

• The Cylinder: The cylinder takes up

          $\frac{3}{3}$
        

  of the cylinder.

The calculus operation

        $\int x^2 \to \frac{x^3}{3}$
      

is not an arbitrary algebraic rule; it is the geometric measurement of how "pointy" shapes fill space compared to "blocky" shapes.

Conclusion

The formula

        $\frac{4}{3}\pi r^3$
      

is not a random string of numbers. The 4 comes from doubling the height (2) and doubling the hemisphere (2). The 3 comes from the pyramidal nature of linear growth (

        $\frac{1}{3}$
      

).

Calculus and Geometry are not separate disciplines here. The integral symbol (

        $\int$
      

) is just an instruction to stack slices. When we follow that instruction, we find that the sphere is the physical manifestation of the space left over when you core a cone out of a cylinder.