Is side length & area proportional?

- So let's draw a square. So it's not perfectly drawn but this is a reasonable attempt at a square, and of course one of the things that we know about a square is all of the sides are going to have the same length. So if the length of this side is x, well all of them are going to have, all of them are going to have length x. We also know this is a, these are going to be right angles, that's what makes it a square. Now the question I want to ask, are the lengths of the side of a square proportional to its area? And I encourage you to pause the video and think about it, are the lengths of a side, are the lengths of the sides of a square proportional to the square's area? Well to think about that let's draw, let's have a little table here. So side length, side length and that's going to be given as x, so that's going to be x, and then I'll have another column, I can draw that a little bit neater so let me use this tool, so there you go, and I'm going to make two columns, two columns right over here, and then over here I'm going to put area. Area, and what's the area going to be? Well the area is going to be one of the sides squared, I suppose you could view it as the width times the height, so its going to be x times x or x squared. So let's just pick some values for x and then figure out what the area is going to be. So if x is going to be equal to one, the area's going to be one times one, which is still one. If x is going to be equal to two, the area's going to be two times two, which is equal to four. If x is going to be equal to three, the area's going to be three times three, which is equal to nine, and I could keep going but I think this is enough, these are enough points to think about is the side length proportional to the area, or is the area proportional to the side length? Now one of the ways you can think about proportionality is, are the ratios between the side length and the corresponding area always the same, so we want to look at the ratio between side length and area, or area and side length, as long as they're always constant. Area and side length, here let me do those as two separate, and side length. So let's make an extra column here. Let's make an extra column here, so there we go, and let's look at that ratio, the ratio of area to side, to side length. So here the ratio's one over one, one over one, which equals one. Now that seems reasonable. For this next one the ratio of area over side length is four over two, four over two, which equals two. I don't even have to go to this third one, I can, I can say this ratio is going to be area over side length is going to be equal to three. And notice I get a different value every time. This is not equal to a constant, I don't have a constant ratio between area and side length so that tells us that these two, that the side length and the area are not proportional. Not, not proportional. And if you look at it, it makes a lot of sense that the ratio is actually going to be equal to the actual side length, that it changes depending on the side length and that's because we're squaring things, it's literally, you're literally taking x squared, if, you know, you have x here, this is going to be x squared and the ratio is going to be x squared over x, which of course is always just going to be equal, this is x times x divided by x, that's always going to be equal to x, and we see that right over here. But it's definitely not proportional.