Explaining the relationship between diameter and circumference from a graph

0:00- [Instructor] We're told that Imani found the diameter and circumference for four circular objects. She plotted the measurements in the coordinate plane. All right, so this is one of the objects. It looks like it has a diameter of six centimeters, and then a radio, sorry, a circumference that's a little bit more than six centimeters.

0:18Here there's another one, diameter four centimeters, circumference a little bit more than 12, and then so on and so forth. Which statements help explain the relationship between the diameter and circumference for these circles? So pause this video. Have a go at that before we do this together.

0:36All right, now let's just read each of them. Actually, before I even read these, let's just see if we can see a relationship. Well, if I were to connect the dots here, it looks like, it looks like they all sit on a line that would roughly go through zero. At least that's what it looks like right over here.

0:54It looks like a proportional relationship. One, we go through zero, and every time you have some diameter, you multiply it by some constant to proportionality to get the circumference. And it looks like if to go from two to 6.3, you're multiplying by three point, well, we could, we know exactly.

1:15You're multiplying by three point, looks like 3.15 at least at the amount of precision they've given us. And then once again, it looks like we are multiplying by 3.15 again. And let's see here, we're multiplying by three, not exactly 3.15, but it looks like so times three point something.

1:39I could do the math, but we can, and then here again, it's times three point something, but it looks pretty close. And these are real world measurements that Imani was doing. So they might not be perfect every time, but it looks like they're 3.1 something in every situation.

1:56So let's look at these choices. The points appear to form a straight line showing that the circumference increases at a constant rate as the diameter increases. Yeah, that looks like what I just said. So I like this choice. Oh, they say choose two answers. All right, so it's good we found one.

2:13The ratio of circumference to diameter changes for larger circles. Well, the ratio would be this number right over here. If we're taking the ratio of circumference to diameter, that would be 6.3/2 or 12.6/4 or 22/7 or 28.3/9. And the measurement, it looks like this ratio might change a little bit based on Imani's measurements, but it doesn't really look like they're changing.

2:43The circumference is exactly three times the diameter. No, that's not right; if that was the case, this wouldn't be 6.3, it would be six, this would be 12, this would be 21, this would be 27. So hopefully D is right. The ratio of the circumference to the diameter is about the same for all the circles.

2:59Yeah, I think that's right. It's about the same. There's a little bit of error maybe in Imani's measurements, but they look about the same. And those of you who have studied circles know what that ratio is. If you take the diameter and you multiply by pi, which is approximately 3.14159, and it keeps going on forever, that that actually is the ratio.

3:23So if these measurements were perfect, that constant to proportionality right over here would be exactly pi. You take two, and then this number right over here would be two times pi, two times 3.14159 so on so forth. This would be four times pi, that would be seven pi.

3:43And the reason it's not exactly that is it looks like her measurements might have been, these are real world measurements, so these might not have been perfectly circular, or her measurements might not be exactly right.