# Understanding the Napkin Ring Paradox in Geometry

Written on

## Chapter 1: Introduction to the Napkin Ring Paradox

The Napkin Ring Paradox presents a fascinating concept in geometry: any two spheres, when hollowed out to the same height, possess identical volumes. This principle can be illustrated through the example of two spheres of different sizes—a small apple and a large grapefruit, or even a proton and a planet. By removing the core of each fruit, we create what we term the "napkin ring."

To better understand this, let's visualize a ring of height 2H alongside a sphere with a radius of H. The sphere in this scenario acts as a degenerate ring, where the central hole has a radius of zero.

When we take a cross-section through the centers of both the sphere and the napkin ring, the sphere will display a circular cross-section with a radius of H, while the napkin ring will reveal an annulus.

### Section 1.1: Exploring Volume through Geometry

The Washers Puzzle reveals an unexpected geometric relationship behind the solution.

Let’s denote the radius of the outer circle as R, which corresponds to the radius of the original sphere, and the radius of the inner circle as X, representing the hole cut through the sphere. The area of the remaining circular strip is determined by the difference in the areas of these two circles.

The area related to the sphere's circular section is influenced by its radius, H.

Next, we will examine a vertical cross-section of the napkin ring.

Using the Pythagorean theorem, we can relate H, R, and X.

By combining the equations derived from our analysis, we discover that the areas of both cross-sections—circular and annular—are indeed the same.

## Chapter 2: Detailed Analysis of Cross-Sections

The video titled "The Napkin Ring Problem" delves into the fascinating geometry behind the napkin ring and how the paradox unfolds.

In our exploration of the cross-section, we can also analyze an arbitrary position Y above the center line.

Using Pythagorean principles, we can relate the outer radius of the annulus, A, to R and Y. The inner radius remains consistent with previous calculations.

Substituting the relevant expressions allows us to simplify our findings.

As we explore the circular cross-section of the sphere at position Y, we can denote its radius as S.

With this analysis, we can see that the areas of both cross-sections at any arbitrary position Y are equal.

In summary, the total volume of both shapes—whether it's the sphere or the napkin ring—is the same, which ultimately confirms the Napkin Ring Paradox.

The video "What is the Napkin Ring Paradox? (And How Does it Work?)" provides a comprehensive overview of the paradox and its mathematical implications.

Bob’s your uncle!