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• 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 it shape will change, but it's 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 spheres cross section minus the area of the cylinders cross section gives us the area of the Napkin rings

• 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 spheres cross section and subtract the area of the cylinders 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 cross section

• Square that 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)

• 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

• minus 1/2 the height of the cylinder squared

• Perfect ok 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 spheres 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)

• Ok let's subtract y squared from both sides the radius of the spheres cross-section squared

• equals the radius of the sphere squared minus

• Y squared will 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 cylinders 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 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 spheres 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 is 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 it's 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 as actually what I love about these glasses

• 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

Hey, Vsauce! Michael here! If you core a sphere; that is, remove a cylinder from it, you'll be left with a shape

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# The Napkin Ring Problem

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林宜悉 posted on 2020/03/28
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