The volume of two different spheres would be the same if the heights of their cores are the same.

Here’s a statement you’d never 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. Believe it or not. This is known as the Napkin Ring problem.

Related media: The Napkin Ring Problem

Put A Ring On It

This problem was devised by Japanese mathematician Seki Kowa back in the 17th century, who used to call the cored shaped an “arc ring.” He eventually came up with a geometric proof that later became known as the Napkin Ring problem — that’s the fact that a cored sphere looks like (you guessed it) a napkin.

Here’s the hypothetical problem: when you core a sphere, you’ll end up removing a cylindrical shape section that would look more like a napkin-ring-shape object. It doesn’t matter the size of the sphere, if you do perform the coring process precisely to create a napkin ring of a certain height, every cored ring of that same height would also have the same volume.

For instance, when you core an orange to create a 5-inch tall napkin ring, then core the Earth the same way, you’ll be left with two rings with different diameters — but having the exact same internal volume. Huh?

What? There’s Proof

Image: Popular Mechanics | 3-D Illustration of the Napkin Ring problem

There are two ways of proving this weird phenomenon: with math, and without. Let’s take the easy one first. No math. If the size of the sphere you’re coring is bigger, so is the size of the cylindrical shape you have to remove in order to get the desired ring of the right height. Coring a golf ball to a 2-inch tall ring requires removing a lot less than coring that of a football to the same size. The smaller the sphere, the thicker the desired ring, which is to say, a small napkin ring of any given height has the same volume as a large one of the same height.

Now the math: In order to find the volume of two napkin rings of the same height, you have to formulate some equations, as usual. You’d need the formula of finding the area of a circle, the Pythagorean theorem, and with a little geometry, then you’re good to go. After going through all that math, you’ll have to simplify the resulting equations in one equation that looks something like this:

As you might have guessed about this equation, its not your usual kind of math. In order to solve for the volume of the napkin ring (V), you’ll need to find the height of the napkin ring (h); and as it turns out, the radius of the sphere is canceled out (this is inconsequential to a napkin ring’s volume). Check out this video from the Vsauce YouTube channel for a full comprehensive explanation.

How Does This Apply To My Life?

Not much. But its really fun. Aforementioned, it doesn’t matter the size of the sphere. If you were to core a tennis ball and a football to create an inch ring each, those rings would have the same volume, no matter what. What about a grape fruit and a black hole? The same result. As long as their heights are the same, their volume will be exactly the same. Blow the minds of your friends next time you’re on a picnic when you see a napkin.

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Written by: Nana Kwadwo, Thu, Jul 25, 2019.

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