We've learned a lot about Geometry so far,

but there's one really important geometric shape that we still need to cover,

and that shape is a circle.

Since the invention of the wheel,

circles have been extremely important to all humanity.

Grog make wheel.

Thanks Grog!

In fact, you probably see circles almost everywhere you turn…

But mathematically, what is a circle?

Well in Geometry, a circle is defined as:

the set of all points that are equidistant (or the same distance) from another single point.

And the best way to understand what that means is to see it in action.

So… here's a single point to start with.

Now let's start drawing points that are equidistant from it.

This point is a foot away to the right.

Now let's make another point a foot away but in another direction. Let's say up here.

Now let's make another one, also a foot away, but in another direction. Right here.

Now let's make another, right here

and another, and another, and another, and another...

Wheew… I'm getting tired!

But do you see what's happening?

The more equidistant points we add, the more the pattern looks like a circle.

That's why a circle is defined as the set of points that are equidistant from a center point.

But of course, we usually don't see it as a set of points because there are infinitely many of them,

so they form a continuous circle.

Okay, now let's learn about the parts that make up a circle.

First of all we have the original point that we started with.

That's called the center, or the origin of the circle.

Next, we have the distance that we used to draw all of the equidistant point that form the circle.

That distance is called the radius.

The radius is important because

it's the distance from the center of a circle to ANY other point on the perimeter of that circle.

And even though a circle only has one radius dimension, you can draw as many radius lines as you want to.

Usually you'll only see one radius line drawn since it's the same length no matter where you draw it.

Another important circle dimension is called the diameter.

The diameter is the distance across a circle.

If you start at one point on the circle and then draw a line straight through the center to the other side,

that distance is the diameter.

As you can see, the diameter is really just the same as two radius lines drawn in exactly opposite directions.

So, for any circle, the diameter is always exactly twice as long as the radius.

All of the equidistant points we drew combine to form the perimeter of the circle.

Remember that perimeter is just the distance all the way around a shape.

But because a circle is a special shape, the perimeter of a circle gets a special name.

It's called the circumference.

The circumference is the distance all the way around a circle.

We're going to learn how to calculate the circumference of any circle in the next video.

We'll also learn how to calculate the area of any circle.

But before we can learn those things, we first need to learn about Pi.

Grog make Pie!

Sorry Grog, not that kind of pie.

In math, the word Pi (which is spelled 'P' 'i') refers to a very special number.

In fact, it's so special that it gets its own symbol.

This greek letter here is the symbol for the number Pi.

But... if Pi is just a number, why don't we write it like that?

Why do we use a special symbol for it?

That's a good question.

And I'll get to that in just a minute.

But first, let's learn what Pi really is by seeing how it relates to a circle.

It turns out that Pi is a really a Ratio!

Now if you're not sure what a ratio is, you can watch our video about them.

But basically, a ratio is just a relationship between two numbers that is written like a fraction.

Pi is the ratio of two different distances on a circle.

It's the ratio of the distance around a circle to the distance across a circle.

And what do we call those two distances?

Yep, the circumference and the diameter.

So Pi is the relationship of the circumference to the diameter.

And as you'll see in a minute, because Pi is a ratio,

it's the same number for any circle, no matter how big or small.

Okay, but what number is it?

What's the value of Pi?

Well, to figure that out, have a look at these two circles,

one big and one small.

We're going to imagine that our circles' diameters are flexible, like a piece of string,

and that we can wrap them around the outside edges (circumferences) of the circles.

So for each circle, if we start at the top and wrap the diameter around the circumference,

we see that 1 diameter is not enough to go all the way around.

So, let's get another diameter and keep going where the first diameter stopped.

Hmmm… two diameters still isn't enough to go all the way around.

It looks like we're going to need to get a third diameter and keep going.

Awwww! So close!!

Three diameters is almost enough but it looks like

we're going to need just a little bit more to form a full circumference.

That little bit more turns out to be about 0.14 diameters.

That means that it takes 3.14 diameters to equal one circumference for any circle, big or small.

So the value of Pi is always 3.14.

Well okay… Pi is a little more complicated than that.

3.14 is really just Pi rounded off to two decimal places.

And we actually have to round Pi off because it's a type of number that's called 'irrational'.

An irrational number has decimal digits that never end and never repeat.

Grog confused.

Yes, 'irrational' numbers are confusing,

but seeing some more of Pi's decimal digits will help you understand what I mean.

To be more precise, Pi is 3.141592653589793238…

and the decimal digits keep on going forever without repeating!!

Pretty amazing, huh?

But the good news is that saying Pi is 3.14 is usually close enough for most math problems,

so that's all you really need to memorize.

And that's why we use a symbol for Pi in equations.

We could write Pi with just two decimal places.

Or we could write it with 5 decimal places to be more accurate.

Or, we could write it with hundreds of decimal places to be super accurate.

Or, we could just use the symbol to represent the true value, which is infinitely accurate.

Okay, so in this video, we've learned what a circle is,

and we've learned about the important parts of a circle:

the center, the radius, the diameter and the circumference.

We've also learned about a very special number called Pi.

Pi is the ratio of a circle's circumference to its diameter,

and its value is about 3.14 no matter what size the circle is.

In our next video about circles, we're going to learn how

we can use the number PI to find the circumference and the area of any circle.

And even though there is not much math you can actually practice in this section, don't worry…

there will be lots of practice problems in the next section to make up for it!

Thanks for watching Math Antics and I'll see ya next time.

Mmmm, Grog good at math!

Learn more at www.mathantics.com