Area formula intuition
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.