I know how to calculate the area of a circle.

Circumference

Epicycle

Pi

So what is the circumference?

Epicycle

Of course not.

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

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.

And now in the x direction, this is the x term, my x radius is 3.

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

Start transport from this point of the circle

Select a point on the circle...

Transport a measure on this circle

Select the segment to transport 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

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

OK, that's interesting.

Let's call it area surface.

Actually, never!

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

Even Newton and Leibnitz, who did a lot of calculus in the late 17th century, devised additional methods of calculating Pi.

What about bigger numbers?

Pi's just a number.

He's reciting the digits of pi.

Circles around the Origin.

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

Then I find the center of the circle around which I'm turning, using the exact same math I just gave you.

Since Archimedes, people all over the world have been obsessed with calculating Pi.

So it's just going to be 3 times 2 or 3 plus 3, which is equal to 6 meters.

A good man... draws a circle around himself and cares for those within.

OK, this is a neat property of circles.

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

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.

How do you derive the length of the circumference? I've forgotten.

In fact, calculating Pi was used to test proper functioning of early computers.

And it's not just circles, either.

Then they find out that the diameter is roughly 2 centimeters.

And actually, circles are a special case of an ellipse.

So let's think about that