# The Napkin Ring Problem

published on July 12, 2020

Hey Vsauce Michael hereif you core a spear that is remove acylinder from it you'll be left with ashape called a napkin ring because wellit looks like a napkin ring it's abizarre shape because if two napkinrings have the same height well they'll

Have the same volume regardless of thesize of the spheres they came from thismeans that if you cut equally tallnapkin rings from an orange and from theearth well one could be held in yourhand the other would have the

Circumference of our entire planet butboth would have the same volumeI mentioned this counterintuitive factwhile making a kendama with Adam Savagecheck that video out if you haven't orbetter yet just come see us we're

Bringing brain candy live to 24 newcities this fall it's gonna be busy butright now we're talking about balls andcoring them I have here two napkin ringsfrom very differently sized spheres oneis from a tiny ball just a little tomato

That I've chord so it's got a littlehole in it right there the other napkinring is made from an orange but bothnapkin rings have the same height thetomato has a smaller circumference thanthe orange which means less volume but

It's ring is thicker which means morevolume both of those effects exactlycancel out so these two napkin ringshave identical volumes they take up thesame amount of space by the way orangeoil is flammable to see why the napkin

Ring problem is true let's discussCavalier ease principle it states thatfor any two solids like these twocylinders I've built here sandwichedbetween parallel planes if any otherparallel plane intersects both in

Regions of equal area no matter whereit's taken from well then the solidshave the same volume now that's clearlytrue here these cylinders are built outof stacks of vsauce stickers 100 in eachstack so their volumes are the same

If I skew one of them like this itsshape will change but it's volume hasn'tit still contains the same amount ofstuff I haven't added or subtractedstickers and Cavalier ease principleensures that they still have the same

Volume because any cross-section takenfrom down here up here in the middleanywherewill always give us a region of the samearea as the other because those regionsare always equal area circles now let's

Apply calgary's principle to napkinrings we can see the two napkin ringswith similar Heights have identicalvolumes by showing that when cut by aplane the area of one's cross sectionalways equals the area of the others now

To do this notice that the area of thespheres cross section minus the area ofthe cylinders cross section gives us thearea of the napkin rings cross sectiondepending on where we slice the napkinring the cross sections will have

Different areas but they will always bethe same as each other let's calculatethe areas of these blue rings first ofall let's call the height of the napkinring H and the radius of the spherethey're cut from capital R all right

Perfect now a cross section of a spherelike this and a cross section of acylinder like this are both circles sotheir areas can be determined by usingpi times the radius squared so if wewant to find the area of the spheres

Cross section and subtract the area ofthe cylinders cross section draw apicture of a cylinder here all we needto do is take pi multiply it by theradius of the sphere cross sectionsquare that and then subtract pi times

The radius of the cylinder squared butwhat are their radii well if this is thecenter of the sphere we can draw a linestraight up to the corner of thecylinder down the side of the cylinderand then connect to form a right

Triangle the Pythagorean theorem willreally help us here it tells us that thelength of one side squared plus thelength of the other side squared equalsthe length of the hypotenuse squaredthis distance right here this side of

The triangle is what we want it's theradius of the cylinder so we'll callthis the little R radius of the cylinderbeautiful little picture there so theradius of the cylinder squared plus thisside length which is just half the

Height of the cylinder so the height ofthe cylinder divided by 2 squared equalsthe hypotenuse squaredthe hypotenuse happens to be the radiusof the sphere itself which is capital Rperfect now let's solve for the radius

Of the cylinder which is what we wantwe'll just subtract H over 2 squaredfrom both sides that'll give us theradius of the cylinder squared equalingthe radius of the sphere squared minus1/2 the height of the cylinder squared

We can take the square root of bothsides so that we wind up with the radiusof the cylinder equaling the square rootof the radius of the sphere minus 1/2the height of the cylinder squaredperfect ok now let's take a look at the

Area of a cross-section of the spherenow for this let's draw a straight linefrom the center out to the edge of thespheres cross-section and then we'll gostraight down and connect back up a lookanother right triangle let's call this

Height Y and notice that this distancenow the side of the triangle down hereis actually the radius of the circlecross section up here they're both equalso we don't want to solve for this theradius of the circle that is the spheres

Cross section okay so we know that theradius of the spheres cross sectionsquared plus this distance squared whichis y equals the hypotenuse squared wellwhat do you know the hypotenuse is theradius of the sphere again capital R

Okay let's subtract Y squared from bothsides the radius of the spheres crosssection squared equals the radius of thesphere squared- y-squared we'll take the square rootof both sides and end up learning that

The radius of the spheres cross-sectionequals the square root of the radius ofthe sphere squared – y squared y is theheight that this cross-section is takenfrom above the equator the higher up wetake these cross sections of the sphere

The smaller their radii will be whereasthe cylinders radius is always the sameno matter where we cut from Oh anywaylet's take these two radii and plug theminto our formulaokay the area of the cross section of

The sphere is what we want first okaythat's just the square root of R squared- y squared not too bad now the radiusof the cylinder is the square root of Rsquared minus 1/2 the height of thecylinder squared now what you might

Notice is that we're taking the squareroot of something and then squaring itso these actually cancel each other outperfect much more simple looking but nowlet's distribute PI to the terms insidethe parentheses so PI times R squared

Gives us PI R squared pi times negativeY squared gives us negative PI y squaredthen a negative pi times R squared isnegative PI R squared negative pi timesnegative H over 2 squared is positive PIH over 2 squared great now we can keep

Simplifying but what you might notice isthat we have a PI R squared and a minusPI R squared let it equal 0 so thesecompletely cancel each other outbut what we're left with are termscontaining no mention of the spheres

Radius whether the radius is large orsmall doesn't matter all you need toknow to find the area of the crosssection of a napkin ring is the heightof the napkin ring why of course isbounded by the height of the napkin ring

These blue areas have the same area aseach other and this will be true nomatter where we cut the cross sectionacross the napkin ring meaning bycavalier YZ principle that both napkinrings have the same volume

Yay with teethbut what does this mean for you for lifein the universe well as we know if youlike it you should put a ring on it butif you like it don't know it's fingerwidth and only want to offer it a

Predetermined amount of material youshould put a napkin ring on it and asalways thanks for watchingon August 21st 2017 there will be atotal solar eclipse the shadow of themoon will race across the contiguous

United States it's going to beincredible and a little bit scary I'msure I will be viewing it from organwith my friends at Atlas obscuraI can't wait but keep your eyes safe ifyou want to view the Eclipse you have to

Have special eye protection theCuriosity box comes with such glassesthese block 99999% of visible lightthat's what it takes to be able to lookright at the Sun that's actually what Ilove about these glasses if there's no

Eclipse going on you can still just lookat the Sun notice that it's a ball maybeimagine what kind of napkin ring you'dlike to make it into the currentcuriosity box is my favorite the onethat you'll get it you subscribe right

Now comes with all kinds of cool stuffthat comes with a poster showing thatall the planets and Pluto can fitbetween the earth and the moonit also comes with science gadgets likethese levitating magnetic rings pretty

Cool also a portion of all proceeds goto Alzheimer's research so it's good foryour brain and everyone else's braincheck it out I hope to see you at braincandy live and as always thanks forwatching

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