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)
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
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