The Napkin Ring Problem

Hey, Vsauce! Michael here! If you core a sphere; that is, remove a cylinder from it, you'll be left with a shape called a Napkin ring because, well, it looks like a napkin ring! It's a bizarre shape because if two Napkin rings have the same height, well they'll have the same volume regardless of the size of the spheres they came from! (Cool) This means that if you cut equally tall napkin rings from an orange and from the Earth, well, one could be held in your hand. The other would have the circumference of our entire planet, but both would have the same volume...

I mentioned this counterintuitive fact while making a Kendama with Adam Savage. Check that video out if you haven't, or better yet, just come see us. We're bringing brain candy live to 24 new cities this fall. It's going to be busy, but right now, We're talking about balls and coring them!

I have here TWO napkin rings from very differently sized spheres; one is from a tiny ball, just a little tomato that I've cored, so it's got a little hole in it right there. The other Napkin ring is made from an orange, but both Napkin rings have the same height. The tomato has a smaller circumference than the orange, which means less volume, but its ring is thicker, which means more volume. Both of those effects exactly cancel out. So these two napkin rings have identical volumes; they take up the same amount of space.

By the way, orange oil is flammable. To see why the napkin ring problem is true, let's discuss Cavalieri's principle. It states that for any two solids, like these two cylinders I've built here sandwiched between parallel planes, if any other parallel plane intersects both in regions of equal area, no matter where it's taken from, well then the solids have the same volume. That's clearly true here; these cylinders are built out of stacks of VSauce stickers, 100 in each Stack, so their volumes are the same.

If I skew one of them like this, its shape will change, but its volume hasn't; it's still contained the same amount of stuff. I haven't added or subtracted stickers, and Cavalieri's principle ensures that they still have the same volume because any cross-section taken from down here, up here, in the middle, anywhere will always give us a region of the same area as the other because those regions are always equal area circles.

Now let's apply Cavalieri's principle to Napkin Rings. We can see that two Napkin Rings with similar heights have identical volumes by showing that when cut by a plane, the area of one's cross-section always equals the area of the others. Now, to do this, notice that the area of the sphere's cross-section minus the area of the cylinder's cross-section gives us the area of the Napkin ring's cross-section.

Depending on where we slice the Napkin ring, the cross-sections will have different areas, but they will always be the same as each other. Let's calculate the areas of these blue rings. First of all, let's call the height of the Napkin ring h and the radius of the sphere they're cut from capital R.

Alright, perfect! Now a cross-section of a sphere like this and a cross-section of a cylinder like this are both circles. So their areas can be determined by using Pi times the radius squared. So if we want to find the area of the sphere's cross-section and subtract the area of the cylinder's cross-section (I'll draw a picture of a cylinder here), all we need to do is take Pi, multiply it by the radius of the sphere's cross-section squared, and then subtract Pi times the radius of the cylinder squared.

But what are their radii? Well, if this is the center of the sphere, we can draw a line straight up to the corner of the cylinder, down the side of the cylinder, and then connect to form a right triangle. The Pythagorean Theorem will really help us here; it tells us that the length of one side squared plus the length of the other side squared equals the length of the hypotenuse squared.

Now this distance right here, this side of the triangle, what we want, it's the radius of the cylinder, so we'll call this the little r radius of the cylinder (beautiful little picture there). So the radius of the cylinder squared plus this side length, which is just half the height of the cylinder, so the height of the cylinder divided by 2 squared equals the hypotenuse squared. The hypotenuse happens to be the radius of the sphere itself, which is capital R.

Perfect! Now let's solve for the radius of the cylinder, which is what we want. We'll just subtract h over 2 squared from both sides; that'll give us the radius of the cylinder squared equaling the radius of the sphere squared minus half the height of the cylinder squared. We can take the square root of both sides so that we wind up with the radius of the cylinder equaling the square root of the radius of the sphere squared minus 1/2 the height of the cylinder squared.

Perfect! Okay, now let's take a look at the area of a cross-section of the sphere. Now, for this, let's draw a straight line from the center out to the edge of the sphere's cross-section, and we'll go straight down and connect back up. Hey, look! Another right triangle! Let's call this height y.

And notice that this distance now, the side of the triangle down here, is actually the radius of the circle cross-section up here. They're both equal, so we even want to solve for this; the radius of the circle that is the sphere's cross-section. Okay, so we know that the radius of the sphere's cross-section squared plus this distance squared (which is y) equals the hypotenuse squared. Well, what do you know? The hypotenuse is the radius of the sphere again (capital R).

Okay, let's subtract y squared from both sides; the radius of the sphere's cross-section squared equals the radius of the sphere squared minus y squared. We'll take the square root of both sides and end up learning that the radius of the sphere's cross-section equals the square root of the radius of the sphere squared minus y squared. Y is the height that this cross-section is taken from above the equator.

The higher up we take these cross-sections of the sphere, the smaller their radii will be, whereas the cylinder's radius is always the same no matter where we cut from. Anyway, let's take these two radii and plug them into our formula.

Okay, the area of the cross-section of the sphere is what we want first. Okay, that's just the square root of R squared minus y squared. Not too bad! Now the radius of the cylinder is the square root of R squared minus half the height of the cylinder squared.

Now what you might notice is that we're taking the square root of something and then squaring it. These actually cancel each other out. Perfect! Much more simple looking. But now let's distribute Pi to the terms inside the parentheses. So Pi times R squared gives us Pi R squared.

Pi times negative y squared gives us negative Pi y squared. Then a negative Pi times R squared is negative Pi R squared, and negative Pi times negative h over 2 squared is positive Pi h over 2 squared. Great! Now we can keep simplifying, but what you might notice is that we have a pi r squared and a minus Pi R squared. Well, that equals 0, so these completely cancel each other out.

But what we're left with are terms containing no mention of the sphere's radius. Whether the radius is large or small doesn't matter; all you need to know to find the area of the cross-section of a napkin ring is the height of the Napkin Ring. Y, of course, is bounded by the height of a napkin ring. These blue areas have the same area as each other, and this will be true no matter where we cut the cross-section across the napkin ring, meaning by Cavalieri's principle that both Napkin Rings have the same volume.

Yay :3 (with teeth) But what does this mean for you, for life in the universe? Well, as we know, if you like it, you should put a ring on it, but if you like it, don't know its finger width and only want to offer it a predetermined amount of material, you should put a napkin ring on it.

And as always, thanks for watching! :D On August 21st, 2017, there will be a total solar eclipse. The shadow of the Moon will race across the contiguous United States. It's going to be incredible and a little bit scary, I'm sure. I will be viewing it from Oregon with my friends at Atlas Obscura. I can't wait, but keep your eyes safe.

If you want to view the eclipse, you have to have special eye protection. The Curiosity Box comes with such glasses; these block 99.999% of visible light. That's what it takes to be able to look right at the sun. Actually, what I love about these glasses is that there's no eclipse going on. You can still just look at the sun, notice that it's a ball, maybe imagine what kind of Napkin ring you'd like to make it into.

The current Curiosity Box is my favorite. The one that you'll get if you subscribe right now comes with all kinds of cool stuff; that comes with a poster showing that all the planets and Pluto can fit between the Earth and the Moon. It also comes with science gadgets like these levitating magnetic rings. Pretty cool!

Also, a portion of all proceeds go to Alzheimer’s research, so it’s good for your brain and everyone else's brain. Check it out! I hope to see you at Brain Candy live, and as always, thanks for watching!