Pi

Factual and interesting representation of how pi helps figure out the circumcfrence of a circle.

what's wrong with the colors?Nnfolz 03:17, 15 August 2006 (UTC)[reply]
Comment: jjron, the area you describe is Pi (units2). That's like saying 2 isn't a number because it doesn't have units or dimensions. 1+1=2. 1x2=2. Or, we can specify units. 1 unit + 1 unit = 2 units. A rectangle with a long side of 2 units and a short side of 1 unit has an area of 2 units2. A rectangle with a long side equivalent in length to the circumference of a circle and a short side equivalent in length to the same circle's diameter has an area of Pi units2.
Pi is more than just the ratio of the circumference of a circle to its diameter -- though that is the most famous definition and the one first taught. But just as "2" is more than just 1+1 (it's also the square root of 4; 18 divided by 9; etc), Pi can also be described other ways. For example, it is the ratio of the area of a circle to the area outside the same circle but within a square with sides of one circle diameter length. (This would be easier to imagine with a picture -- draw a square on a piece of paper, and then draw a circle within it that touches all four sides. The ratio of the area within the circle to the area outside the circle but within the square is Pi.)
Area in circle: пr^2. Area in square: 4r^2. Area outside circle but in square: 4r^2 - пr^2. How do you get п from any ratio?? — BRIAN0918 • 2006-08-17 19:03
Hazy and apparently incorrect recollection of a programming project I did in college 17 years ago. Whoops. It's the ratio of a circle's area to the area of a square with sides the same length as the circle's radius. My first example was bad, but the point stands... Pi can be defined by methods other than the ratio of the circumference of a circle to its diameter. -- Moondigger 19:52, 17 August 2006 (UTC)[reply]
Back to the animation we've been discussing. If the point is to demonstrate the ratio of a circle's circumference to its diameter, specifying the area of the rectangle laid out in the animation is superfluous and potentially confusing. That it lacks the units2 designation is potentially even more confusing, though it's no more 'incorrect' than if somebody made an animation showing a rectangle twice as long on one side as the other, and calling the area "2." (The units2 is understood.) In any case, I agree this animation shouldn't even bother specifying the area. -- Moondigger 13:33, 17 August 2006 (UTC)[reply]
Eh, I didn't think it did specify the area.. though I suppose we'd have to ask the author. But the area is as much Pi as it could be 3 if the box ended a bit earlier. --Gmaxwell 19:34, 17 August 2006 (UTC)[reply]
But if the box 'ended earlier,' it wouldn't have a side that's equivalent to the circumference of the circle, with the circle's diameter as its height. -- Moondigger 19:55, 17 August 2006 (UTC)[reply]
That comment of mine was to address the complaint that unitlessness was a problem, I wasn't suggesting a change. I wouldn't change anything about the image. --Gmaxwell 19:59, 17 August 2006 (UTC)[reply]
Ah, I understand now -- I agree that unitlessness is not a problem from a technical perspective, since the "units2" is generally understood. But I do think demonstrating the area in the 'growing pink rectangle' is ill-advised from a pedagogical perspective. -- Moondigger 20:31, 17 August 2006 (UTC)[reply]
Sigh. Firstly, Moondigger, that stuff you were saying about 2 was the strangest thing I've ever read from you - if it made any sense it could only be that you totally misinterpreted my comment.
Now, an analogy for those who think π and π units2 are the same thing. I ask you to do a job and say I'll pay you $100 for it. After you've done the job I give you a scrap of paper with the number 100 written on it. Now are you going to be happy with my payment? Unlikely. But when you argue that π and π units2 are the same thing, you are making the same mistake. π is not the same as π units2 (just as the $100 is different from the number 100). You see, the units do make a difference. The number is the same, but the thing is not.
When you also say that "the units2 is generally understood", that is also not valid. Let's be realistic; is this graphic aimed at people that understand the concept of Pi intimately, or those learning the concept? It might be fine for you and me, who can conceptualise it, but what about the 13yo kid this is presumably aimed at as a learning aid? If you don't think it will be a problem for 99.9% of them then I'd say you've obviously never tried to teach this sort of thing. And if you're happy to teach the concept of area without reference to units, then please don't ever try teaching it as that would be an unfortunate experience for any hypothetical students.
If this is so confusing for people who presumably already understand π, then it would surely be even moreso for students who would henceforth associate π with an area. In fact what I'm getting from a number of the comments is that there are several voters that don't understand this fully, and probably don't understand one or more of that (i) π is a constant and not an area, (ii) the area of the rectangle shown is not same as the area of the circle shown, and therefore (iii) that the area of a circle is not π (or π units2), i.e., to simplify, using this many students could interpret that π is the area of a circle (and most likely any circle). And if you knocked that out of them, then they'd still think π was some sort of area, so then you'd have to unteach that, and so on - doesn't seem very effective.
I'm not totally convinced the author even intended to equate π with the area, and I'm sure they didn't intend this confusion. And all because of that darn rectangle. --jjron 08:10, 18 August 2006 (UTC)[reply]
First, I think it's clear based on the way the pink box "grows" that the author did intend to show that the area of a rectangle with length equivalent to the circumference of a circle and height equivalent to the diameter of the circle is Pi, and that the units2 is understood but unstated. Nevertheless, I agree that it is potentially confusing and shouldn't be a part of this animation, and have agreed on that point from the start.
As for the rest of it, I'm not sure why you didn't follow what I was saying. The basic gist of it is that like 2, Pi is a number. 2 can be defined many ways - 1+1, 18/9, etc. Ditto for Pi. One definition is "the ratio of the circumference of a circle to its diameter." That's the classic definition. Another is "the ratio of the area of a circle to the area of a square with sides the same length as the circle's radius." There are other definitions as well.
I don't think your analogy about being paid $100 for a job was directed at me, but I will say this. Dollars, like units2, are often unstated but understood. To wit:
"How much are you paying for this job?" asked Harold.
"A hundred," replied Jim.
Now if Harold does the job and Jim attempts to hand him a piece of paper with the number "100" written on it for payment, Jim risks getting a punch in the nose. The units, dollars, were understood from the start even though they were not stated. I believe that's the same thing going on in this animation.
When children are taught the formulae for finding area, at least at the school I attended and the one my children are now attending, the units are seldom mentioned, because they are understood even if they're not stated. For example, they are taught that the area of a rectangle is length x width. The area of a circle is Pi times r2. Nobody says the area of a rectangle is length units times height units, or that the area of a circle is Pi units times radius units squared. Yet you can't get units2 in the answer unless the units were understood from the start to be attached to the length, width, radius, or Pi. They are present but unstated. This is all a side issue anyway, since it relates to the pink rectangle and we both agree that it shouldn't be in the animation in the first place. Given that we agree it should be eliminated, I see little point in discussing units2 further.-- Moondigger 12:23, 18 August 2006 (UTC)[reply]

Comment There is another, somewhat similar animation on the Pi article, shown here:

But the article is screwed up. The text description next to this animation is the correct description for what happens in the wagon-wheel demonstration nominated above. Somebody must have moved the images around without updating the text. In any case, neither animation is a worthy FP, IMO. The wagon-wheel suffers from trying to depict too much; this one doesn't show enough (like a horizontal scale). -- Moondigger 18:34, 18 August 2006 (UTC)[reply]

Not promoted But it's clear from the comments that a similar animation addressing the concerns raised above would likely pass FP muster. -- Moondigger 01:02, 25 August 2006 (UTC)[reply]

Category