[music] We know what the area of a rectangle is.
Well, if you've got a rectangle, say, of width a and height b, what should the area
of this rectangle be? The area of this rectangle is a times b.
So what's the area of a triangle? Well, if we've got a, triangle say of
height b and width a, right. The area of this triangle is 1 half ab.
But why? Why is that the area of a triangle?
Well this is really what the area of the triangle has to be in order to be
consistent with the area of the whole rectangle.
I mean, look, I could take two of these triangles.
Right here's two triangles, both have width a and height b.
But if I push those 2 triangles together, then I've got the a by b rectangle.
So the area of these 2 triangles together has to be the area of the rectangle a
times b. But each of these triangles has half the
area of the rectangle. So the area of just the triangle should be
half the area of the rectangle. Another way to say this is just that, you
know, areas sort of add. You know, area of some figure plus area of
some other figure. Is equal to the area of those two things
together. Right?
In this particular case, the area of this triangle plus the area of this triangle,
is the area of these two things together, which is the area of a rectangle.
You can play this kind of game to relate the areas of different figures that
supposedly have the same area. For example, I could start with a
rectangle of width 2 and height 1. Or, I could start with a square whose side
length is the square root of 2. Now, both of these figures have the same
area, right? The area in both of these figures is 2.
And you might ask, well, you know, if the whole deal with areas is just so that they
sort of cut up and add appropriately. Is it possible for me to cut up this
figure in order to get this figure? And, and yeah, it is in fact possible.
Right? If I cut along the diagonals here, these 4
pieces can be rearranged into these 4 pieces here.
So, and very visibly, right, if you believe that, you know, sort of areas add
in this way that I'm allowed to rearrange pieces.
Then you might believe that these 2 things have the same area, regardless of whether
you do the calculation or not, right? These 2 pieces have the same area because
I can take this thing and cut it up and get this thing.
Well here's a challenge for you right along these lines.
For instance can you take a 3x1 rectangle, and a square of side length root 3.
And visibly show that these things have the same area.
I mean yes. The calculation, right, is telling me that
they both have the same area of 3, 1 times 3 is the same as the square root of 3
times the square root of 3. But what I'm looking for is some sort of
geometric proof where you cut up this thing and rearrange those pieces to build
this thing, right. That's your challenge.
So this is really fascinating, right. What is area?
Area is a number, but that number captures this geometric idea that if two things
have the same number, the same area, then you can take the one thing and chop it up
into pieces to get the other thing. And yet that perspective is totally
destroyed as soon as we start thinking about something with a curved boundary,
like a circle. So here's a unit circle by which I mean
the radius has length 1 unit. In the area of the unit circle is pi
square units. I mean yes we know what the formula for
the area of a circle says right, it says that the area of that unit circle Is pi
square units. But what does that mean?
What does it mean to say the circle's area is pi square units?
Well I'll tell you what it doesn't mean. It doesn't mean that I can start with a
rectangle of area pi square units. So say a rectangle of height 1.
And with pi, right? To say that the area of this circle is
equal to pi does not mean that if I just start with a rectangle with that same
area, that I can necessarily make some finite number of cuts to this rectangle.
And take those pieces and rearrange them to form this disc.
All right. There's really two competing concepts
here. You've already got an idea of area, you
know? Because you've seen area before.
And this idea of area is really based on this idea that two things have the same
area if you can cut the one thing up and rearrange the pieces to make the other
thing. But as soon as you're dealing with objects
with curved boundaries, say, you've suddenly got a problem, because it's not
going to be possible for you to take this object and just chop it up and produce
this nice smoothly curved object. So, somehow you've gotta refine what area
even means. When I say that the area of a circle is pi
square units, I'm secretly speaking in the language of limits.
So, when I say that this circle has area pi, I mean that if I say cut it up into
small enough little rectangles. The area of these rectangles, which
actually makes sense, because I know how to calculate the areas of rectangles, by
multiplying their widths times their heights.
So, if I add up the areas of these rectangles, I get an answer that's as
close as I'd like to the number pi. And I can get the answer as close as I
want, provided I make these rectangles thin enough.
And it's only in that sense that it, it even means anything to say, that the area
of the circle is pi. It's in the sense of that limit.
So even defining area requires talking about limits, which I think is remarkable.
I mean we learn about area in elementary school, right?
We never talk about limits in elementary school.
And yet somehow there's a real depth, a real subtlety to even this concept of
area, you know? And I think that's really exciting.
Right? It suggests that there's something
interesting yet to be learned even about concepts that we think we understand
really well.