Area of a circle

Area of a circle and how it relates to radius and diameter

So we've got a circle here--
doesn't look like a perfect

circle, but we can use our
imaginations --and let's say

it's got a radius of 3 meters.

My question, or the question
we're going to answer in this

video is what is the
area of this circle?

And remember, the area is just
how much space this circle

takes up on a surface, or on
this computer screen that

you're watching, or on
this piece of paper.

If this was a room, it's how
much carpeting you would need

to fill out this circular room.

That's what the area is.

Now, I'm not going to prove it
to you, and we'll do that

later, but the area for circle
just takes on a fairly

straightforward formula and I
want to just get you used to

applying that formula.

So the area of a circle
is equal to pi.

Remember, pi was that number
that people figured out was the

ratio between the circumference
and the diameter of the circle.

It's 3.14159, keeps
going on and on and on.

It's just the number, but
it's a very magical number.

Pi times the radius squared.

In fact another way of defining
pi:-- you could even rewrite

this right here --the area over
your radius squared- so

this is your radius.

If you multiply the radius
times itself you could imagine

that would be the area of a
cube that's like that --that

the ratio between the area of
this entire circle and the

ratio of this cube right
here-- or this square.

I shouldn't say a cube.

Cube would be if we went into
3D --but the ratio of the area

of the circle to this square
right here is also

equal to pie.

That could be actually
an alternate way of

defining what pi is.

And if you were to measure it
very carefully using-- there's

thousands of methods you could
do it --you would get 3.14159

and keep going on
and on and on.

But we're not going to delve
too deeply into that.

Maybe one day I'll make a
whole play list on pi.

But we just need to know that
the area is equal to pi times

r squared, so let's just
apply the numbers here.

So in our example, the area is
equal to pi times 3 meters

squared, which is equal to pi
times 9 meters squared, or the

conventional way to write this
is, equal to 9pi

meters squared.

And remembered 9pi, the
convention is just to leave it

that way, but this is the same
thing as 9 times 3.14159, which

is probably going to be, like,
28 point something

meters squared.

Just remember, this is just
some number and it's not 9.

It's actually closer to 28
because it's going to be

9 times 3.14159, but we
just leave it like that.

And that normally will be
good enough for you to

say, hey that is my area.

That's my area: 9pi.

Now let's go the other way:
let's say I have a circle and

let's say that someone would
say that the area

is equal to 16pi.

What is the diameter of
that circle going to be?

Well, we know that area
is equal to pi times

the radius squared.

So at least let's
figure out the radius.

So the area, 16pi, is equal to
pi times our radius squared.

I'm just applying this formula.

We're just going to keep
applying this formula

over and over again when
we're dealing with area.

So area, which we've been
told is 16pi, is equal to

pi times radius squared.

Now, if we divide both sides of
this equation by pi, we get

16 is equal to r squared.

And then you take the square
root of both sides and

you get 4 is equal to r.

I guess r could also be equal
to negative 4, but we're

dealing with distances here;
you can't have a

negative radius.

Or at least in the world
we're living in right now.

Just keep things simple;
we just want to keep our

distances positives.

So let's say that this
has a radius of 4.

Now if the radius is 4,
what is its diameter?

Well, the diameter is always
going to be 2 times the radius.

So this 4, we're going to
have another 4 there.

The diameter is equal to 8.

Now let's do a slightly harder
one that will kind of compound

some of the other things that
we've learned in the past.

So let's say that I
have a circle here.

Let's say that its
circumference is equal to 20pi,

and I want to know its area.

So way you do all these
problems is just figure out

everything you can, given what
they've given you, and then

maybe you can work out the
thing they're asking for.

So if I know that the
circumference is 25, what do

I know about its radius?

Well, we saw in the last video
that the circumference is equal

to 2pi times the radius.

So if the circumference is
equal to 20pi, we could write

that 20pi is the circumference
is equal to 2pi

times the radius.

Now if you divide both sides of
this by pi, those cancel out.

Then if you divide both sides
by 2, this becomes a 1, this

becomes a 10, or you get
the radius is equal to 10.

Which makes sense, right?

2pi times 10 is going
to be equal to 20pi.

So we've figured
out our radius.

Now, we know that the area is
equal to pi times r squared.

And lucky for us, using the
circumference, we were able

to figure out the radius.

Now using the radius, we
can figure out the area.

So the area is going to be
equal to pi times r squared,--

r is 10 --times 10 squared,
which is equal to pi times 100.

Or it's equal to 100pi.

Just like that.

so your circumference was 20pi,
when you went around the

circle, but your area of
your circle is 100pi.

And if I gave you units it
would be 100pi units squared.

That is your area
right there: 100pi.

Anyway, I think that's pretty
good initial exposure to

the area of a circle.

I'll see in the next video.