The Napkin Ring Paradox

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Essentially, when you cut a hole of a certain diameter in a sphere, you are left with a ring that resembles a napkin ring, thus the name.

The perplexing aspect of this problem is when you try to find the volume of this ring, a series of algebraic cancellations occur and the final volume is found to be the same regardless of the original sphere’s radius. Essentially, a napkin ring cut from a sphere of a given size has the same volume as another napkin ring cut from a differently sized sphere, as long as the ring height is the same.

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Your mind might not be blowing yet, but think about the implications. If you core an orange to create a napkin ring that is 2 inches tall, it would have the same volume as if you cored the earth to create a napkin ring that was 2 inches tall. This sounds absolutely impossible, but if you check the math, you’d find that it is very true.

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