Select the segment to transport on the circle...

Circumference

Epicycle

Pi

So what is the circumference?

And it's not just circles, either.

Epicycle

I know how to calculate the area of a circle.

Of course not.

Do it mathematically correct.

Transport the measure of a segment or arc over a line or circle.

Maybe it's 3 meters put some units in there.

So if you were to get your tape measure out and you were to measure around the circle like that, what's that distance?

And I were to tell you that its circumference is equal to 10 meters that's the circumference of the circle.

Select a point on the circle...

Tangent to This Circle

And so for area we need to know the radius 'cause area's pi R squared, so we can figure out the radius from the circumference by saying, well circumference is equal to two times pi times the radius of the circle, or if you say 36 pi, which we were told this is the circumference, is equal to two times pi times the radius.

The inversion of a point, line or circle with respect to a circle

That's going to be the exact same thing as the circumference of either the top or the bottom of the cylinder

I can draw a neater circle than that. So let's say the circle loks something like that.

And let's say when their tape measures weren't so good,

OK, that's interesting.

Let's call it area surface.

Actually, never!

My volume is going to be 4 3...And then I don't want to just put a pi there because that might interprit it as 4 divided by 3 pi.

We would call this JK, the two end points of the arc, the two points on the circle that are the inputs of the arc, and you would use a little notation like that, a little curve on top instead of a strait line.

The diameter and radius are easy, they're just straight lines you can measure with a ruler.

Inversion of Point, Line or Circle

Start transport from this point of the circle

What about bigger numbers?

Pi's just a number.

He's reciting the digits of pi.

Osculating circle of this curve

Actually, you probably recognize a few of them already, and I'll write them out.

Transport a measure on this circle

It'll be the circumference of this ring times it's height.

Circle by Center Line

The center of the osculating circle to a curve

Two different mythologies, two different ways of looking at the world.

OK, this is a neat property of circles.

If we do it with radians, we know that each revolution is 2 pi radians

It's that x point that we're evaluating it at.

With that out of the way, let's just apply this radius being 7 centimeters to this formular right over here.

So we're going to have a volume is equal to 4 thirds pi 7. cm 3.

That's where all of the rest of the light in the universe is supposed to go