Puzzles & Quizzes

The Napkin Ring Problem Doesn't Seem Possible, But It Is

Here's a sentence you won't believe: If you core an orange and core the Earth so that the remaining rings are the same height, those rings will also have the same volume. Don't believe us? Read on.

Put a Ring on It

This problem dates back to the 17th century with Japanese mathematician Seki Kowa, who called the cored shape an "arc ring." His geometric proof eventually came to be known as the napkin ring problem, due to the fact that a cored sphere looks a lot like a napkin ring.

When you core a sphere, you end up removing a cylinder-shaped section and you're left with a napkin-ring-shaped object. No matter the size of the sphere, if you perform the procedure to create a napkin ring of a certain height, every ring with that height will have the same volume. For example, when you core an orange to create a 2-inch-tall napkin ring, then core the Earth to create a 2-inch-tall napkin ring, you're left with two rings with vastly different diameters — but the exact same internal volume.

Wait, What?

There are two ways to explain this bizarre phenomenon: with math, and without math. The math-free version goes like this: As the sphere you're coring gets bigger, so does the size of the cylinder you have to remove to get a ring of the right height. Coring a tangerine to get a 2-inch-tall ring requires removing a lot less than coring a giant grapefruit to get a 2-inch-tall ring. The smaller the sphere, the thicker the resulting ring, which is why a small napkin ring of a given height can have the same volume as a larger napkin ring of the same height.

In order to find the volume of each napkin ring with math, you have to formulate some equations that use the area of a circle, a little geometry, and the Pythagorean theorem. What you'll find is that after going through all of the math and simplifying the resulting equations, you're left with an equation that looks like this:

One thing to notice about this equation is that in order to solve for the volume of a napkin ring (V), all you need is the height of the napkin ring (h). It turns out that the radius of the sphere is canceled out — it's ultimately inconsequential to a napkin ring's volume. For a full explanation of how that equation was reached, check out the video from the Vsauce YouTube channel below.

Share the knowledge!

Pushing this problem to the extreme, if you were to core a golf ball and the sun to create two 1-inch rings, those two rings would have the same volume. What about a pea and Jupiter? As long as the napkin rings have the same height, the resultant volume will be exactly the same. Now you can blow your family's minds next time you're at a holiday dinner and the napkin rings get pulled out of the cupboard.

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For more wild math problems, check out "My Best Mathematical and Logic Puzzles" by the late, great Martin Gardner. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

Written by Trevor English July 26, 2018

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