Relationship between Circumference and Area
Date: 11/08/2007 at 09:41:13 From: Channel Subject: shortcut between area and circumference and back Hello, My daughter (grade 6) asked me to find a shortcut for converting from Area to Circumference and back. It seems to me that if you have a round ordinary circle and know the area then radius is a component of that number. I know just enough to be dangerous and she was curious (hooray) so we set about it. Most confusing is that it seems intuitive that there should be a relationship between the two that could be easily defined in an elegant equation without resorting to radius. Also frustrating, I'm questioning my math; algebra, order of operations, canceling terms and such. We calculated circumference and area from 1 - 20 in a spreadsheet to test our theories against. I looked for patterns and found that: Circumference increases at a rate of 6.28 per unit of radius. Area increases at an increasing rate of which 6.28 is a component. I found that A=1/2C*R and C=A/r*2 (not helpful because one needs the radius to solve. My daughter went this way: A = Pi * r * r C = 2 * Pi * r so A/r = Pi * r C/2 = Pi * r so A/r = C/2 Also not helpful because radius is required. I realized that there's an extra squared in the Area formula that we don't use for calculation only for presentation so I thought that might be my problem. The result of A = Pi r r is a number like 3.14 if the radius is 1. However, my answer to the area question for this circle has to be 3.14 cm^2 ... Still haven't figured out how this fits in.
Date: 11/08/2007 at 09:59:43 From: Doctor Ian Subject: Re: shortcut between area and circumference and back Hi Channel, >My daughter (grade 6) asked me to find a shortcut for converting >from Area to Circumference and back. Nice! She's thinking like a mathematician. Why keep solving special cases of the same problem over and over, if you can just solve the general case one time and be done with it? >It seems to me that if you have a round ordinary circle and know the >area then radius is a component of that number. I know just enough >to be dangerous and she was curious (hooray) so we set about it. Good for you. In the long run, she'll probably learn more from your willingness to jump in and play around and look for the answer, than from actually finding the answer itself. It's that kind of playing around that kids are supposed to be learning in their math classes, but it hardly ever works out that way. >Most confusing is that it seems intuitive that there should be a >relationship between the two that could be easily defined in an >elegant equation without resorting to radius. That's a good intuition. >We calculated circumference and area from 1 - 20 in a spreadsheet >to test our theories against. I looked for patterns and found that: >Circumference increases at a rate of 6.28 per unit of radius. And 6.28 is about twice pi. Interesting. >Area increases at an increasing rate of which 6.28 is a component. Again, interesting. >I found that A=1/2C*R and C=A/r*2 (not helpful because one needs the >radius to solve. It might be more helpful than you think. But let's set that aside for now. >My daughter went this way: > >A = Pi * r * r >C = 2 * Pi * r >so >A/r = Pi * r >C/2 = Pi * r >so >A/r = C/2 > >Also not helpful because radius is required. Again, it might be more helpful than you think. Let's go back to where your daughter started: A = pi * r * r C = 2 * pi * r But before we start pushing symbols around, let's think about what we'd like to end up with. We want a formula where we can input an area and get a circumference, or input a circumference and get an area, right? That is, we want an equation that contains ONLY an area and a circumference, with no mention of the radius. Suppose, then, that we solve the first equation FOR the radius: A = pi * r^2 A/pi = r^2 sqrt(A/pi) = r And suppose we solve the second equation, also FOR the radius: C = 2 * pi * r C/(2*pi) = r Now we have two expressions equal to r, neither of which involves r itself. Which is almost what we want. To get what we DO want, we just set those expressions equal to each other: r = r C/(2*pi) = sqrt(A/pi) And this is what we were looking for: an equation that relates A and C directly, with no mention of r. Of course, it's kind of ugly, with that square root in there. So we can square both sides, to get C^2 A ------ = -- 4*pi^2 pi which looks a little better. If we cross-multiply, we get C^2 * pi = 4 * A * pi * pi and since multiplying by pi on both sides doesn't accomplish anything, we can cancel that: C^2 = 4 * A * pi This makes sense, that C would have to be squared, since C is going to have units of length, while A will have units of area, or length^2. Of course, we'll want to check it, with some simple cases, to make sure we haven't made some careless error. If a circle has a radius of 1, it has an area of A = pi * 1^2 = pi and a circumference of C = 2 * pi * 1 = 2*pi So let's substitute those into our equation: C^2 = 4 * A * pi (2*pi)^2 = 4 * pi * pi 4 * pi^2 = 4 * pi^2 So that works, at least for one case. I'll leave it to you to try some others. Note that this doesn't PROVE anything! We might just get lucky (or unlucky!) in choosing just some cases that happen to work. But it's a good way to get some confidence that we're right. All of which is to say, you were really on the right track, but just missing the insight that the way to 'get rid' of r was to solve for it! Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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