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## Area formula intuition

Current time:0:00Total duration:5:20

## Video transcript

I've got three rectangles
here, and I also have their dimensions. I have their height
and their width. And in fact, this one right here
has the same height and width, so this is actually a square. So let's think about how
much space they each take up on my screen. And since we're doing
everything in terms of meters, since all of the
dimensions are in meters, I'm going to measure the area
in terms of square meters. So let's see how
many square meters I can fit onto this
yellow rectangle without going outside
of its boundary and without overlapping. So I can fit 1 square meter. Remember, a square
meter is just a square where its length is 1 meter
and its width is 1 meter. So that's 1 square meter, 2, 3,
4, or 5, and 6 square meters. So we see here that the
area is 6 square meters. Area is equal to
6 square meters. But something might
be jumping out at you. Did I really have to sit
and count 1, 2, 3, 4, 5, 6? You might have recognized
that I could view this as really 2 groups of 3. And let me make that very clear. So, for example, I could
view this as one group of 3 and then another group of 3. Now, how did I get groups of 3? Well, that's because
width here is 3 meters. So I could put 3 square
meters side by side. And how did I get the 2 groups? Well, this has a
length of 2 meters. So another way that I
could have essentially counted these six things
is I could have said, look, I have a
length of 2 meters. So I'm going to
have 2 groups of 3. So I could multiply 2 times
3, 2 of my groups of 3, and I would have gotten 6. And you might say, hey, wait. Is this just a coincidence
that if I took the length and I multiplied
it by the width, that I get the same
thing as its area? And no, it's not, because
when you took the length, you essentially said, well,
how many rows do I have? And then you say when you
multiply it by the width, you're saying, well,
how many of these square meters can I fit into a row? So this is really a
quick way of counting how many of these
square meters you have. So you could say that 2
meters multiplied by 3 meters is equal to 6 square meters. Now, you might say, hey, I'm
not sure if that always applies. Let's see if it applies
to these other rectangles right over here. So based on what
we just saw, let's take the length, 4 meters,
and multiply by the width, and multiply by 2 meters. Now, 4 times 2 is 8. So this should give
us 8 square meters. Let's see if this is
actually the case. So 1, 2, 3, 4, 5--
and you see it's going in the right
direction-- 6, 7, and 8. So the area of this rectangle
is, indeed, 8 square meters. And you could view
this as 4 groups of 2. So you could literally
view this as 4 groups of 2. That's where the 4
times 2 comes from. So you could view it as
4 groups of 2 like this. Or you could view
it as 2 groups of 4, So 1 group of 4 right over here. So you could view this is 2
times 4, and then 2 groups 4. I want to draw it a
little bit cleaner. Now, you could
probably figure out what the area of
this rectangle is. It's actually a
square, because it has the same length
and the same width. We multiply the length, 3
meters, times the width, so times 3 meters, to get 3
times 3 is 9-- 9 square meters. And let's verify it again just
to feel really good about this multiplying the dimensions
of these rectangles. So we have 1, 2, 3,
4, 5, 6, 7, 8, and 9. So it matches up. We figure out how many
square meters can we cover this thing with,
without overlapping, without going over
the boundaries. We get the exact same thing
as if we multiplied 3 times 3, if we multiplied the length
times the width in meters.