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.