The napkin ring that ignores the sphere's size

Question

Take a solid sphere and drill a straight cylindrical hole right through its center, removing the middle and the two spherical caps, leaving a "napkin ring." Suppose the finished ring is exactly 6 units tall (the height of the leftover band).

What is the volume of the ring? You may be missing a number, that's the point.

Accepted Answer

The volume is 36\pi, and remarkably, you don't need the sphere's radius. Every napkin ring of height 6 has the same volume, whether it came from a grapefruit-sized sphere or a planet-sized one. The reveal. For a napkin ring of height h, the volume is V = \frac{\pi h^3}{6}, with no dependence on the sphere's radius R. Plugging in h = 6: V = \frac{\pi \cdot 6^3}{6} = \frac{216\pi}{6} = \mathbf{36\pi}. Why the radius cancels. A bigger sphere forces a wider drill hole to leave the same height 6. The extra sphere material is exactly matched by the extra material bored away, so the leftover ring ha…

The napkin ring that ignores the sphere's size

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