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

Of course not.

Let's say that on this circle they measured around it and they found out that the circumference is 6 centimeters, roughly we have a bad tape measure right then.

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

What about bigger numbers?

Circumference

It's like this I drew it smaller.

So let's say that somebody had some circle over here let's say they had this circle, and the first time with not that good of a tape measure, they measured around the circle and they said hey, it's roughly equal to 3 meters when I go around it.

Epicycle

Pi

Well that was just for this circle, but let's say they measured some other circle here.

So what is the circumference?

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

Epicycle

I know how to calculate the area of a circle.

Now, the next most interesting thing that you might be wondering about a circle is how far is it around the circle?

Do it mathematically correct.

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?

Maybe the ratio of the circumference to the diameters always fixed for any circle.

Select a point on the 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.

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.

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.

Pressure

Measure

In fact, the ratio between the two, the circumference divided by the diameter, will always be the same number, no matter how big or small the circle gets.

Start transport from this point of the circle

Pi's just a number.

He's reciting the digits of pi.

And then they measure the radius of the circle right here or the diameter of that circle, and they'd say oh, the diameter

They say my diameter's definitely 1, but when I measure my circumference a little bit, I realize it's closer to 3.1.

Well, many thousands of years ago, people took their tape measures out and they keep measuring circumferences and radiuses.

Transport a measure on this circle

Select the segment to transport on the circle...

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

Measurements counts

So you're the inventor, the founding mother of mathematics, and you need to define what a to the zero is.

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.