# Questions about pi **URL:** \ **Category:** Factual Questions\ **Created:** [July 6, 2013, 5:17pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796 "2013-07-06T17:17:43Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [July 7, 2013, 2:33am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/21 "2013-07-07T02:33:38Z")
> [@deltasigma](#): > > edit: so in that sense, a circle has an infinite number of “sides”. Yes, in several senses, a circle is the limit of a regular polygon as the sides go to infinity. This property is sometimes used for example in physics when dealing with circular paths. ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 2:37am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/22 "2013-07-07T02:37:21Z")
> [@Asympotically\_fat](#): > > Of course a circle doesn’t have sides - y’know I do realize that, but in no sense I can think of off the top of my head is it like a a regular polygon of pi sides. > [@Asympotically\_fat](#): > > Yes, in several senses, a circle is the limit of a regular polygon as the sides go to infinity. This property is sometimes used for example in physics when dealing with circular paths. I was just trying to encourage some non-linear thought but I guess more often than not that tends to come off the wrong way. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 3:51am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/23 "2013-07-07T03:51:02Z")
I like non-linear thought. However, let’s see where this actually takes us: > [@deltasigma](#): > > Can’t you think of pi as how many sides a circle has. If you inscribe a circle inside of a square so that 2r = the length of one side then > > 1. the circumference of the circle is 2 \* pi \* r and that of the square is 4 \* 2 \* r > > 2. the area of the circle is pi \* r^2 and the area of the square is 4 \* r^2 (also = (2r)^2) > > 3. The volume of a cylinder is height \* pi \* r^2. > > So to continue the example let’s assume a height of 2r. > > The volume of the cylinder would be 2r \* pi \* r^2 = 2pi \* r^3 while the volume of the cube would be 4 \* 2r \* r^2 = 8 \* r^3 (also = (2r)^3) Example #3 is just the same as example #1, but multiplied through by height. (It would be more interesting if it were comparing spheres and cubes rather than cylinders and cubes, but then, it wouldn’t actually work out…) So it adds nothing new to account for. Instead, let’s ask ourselves why #1 and #2 line up in this way. Well, with the language of calculus under your belt, you can observe that example #2 is just the same as example #1 but integrated (put another way, example #1 is just the same as example #2 but differentiated). For any family of figures parameterized by r, if increasing r at a particular rate means pulling each bit of the surface out perpendicularly to itself at that same rate, then we’ll find that the rate at which interior is added is simply the total, over each bit of the surface, of the size of that bit times the rate at which r is increasing. Which is to say, the rate at which interior is added over the rate at which r is increasing is the total size of the surface. In calculus jargon, the surface size is the derivative of the interior size, as a function of r. So the role of 4 and pi is not to count number of sides; rather, their role is simply to measure the area when r = 1. For any 2d figure given as a function of r in this way [in particular, regular polygons and their limiting case as a circle], we’ll have an area formula of the form kr[sup]2[/sup] and a perimeter formula of the form 2kr, for the same k. Similarly, for any 3d figure of this sort, we’ll have a volume formula of the form kr[sup]3[/sup] and a perimeter formula of the form 3kr for the same k. And, in general, for an n-dimensional figure, we’ll have an interior formula of the form kr[sup]n[/sup] and a surface formula of the form nkr[sup]n - 1[/sup]. But, again, k doesn’t track the number of sides. k just tells us the interior size when r = 1. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 4:01am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/24 "2013-07-07T04:01:23Z")
> [@Indistinguishable](#): > > Similarly, for any 3d figure of this sort, we’ll have a volume formula of the form kr[sup]3[/sup] and **a perimeter formula of the form 3kr** for the same k. Er, this should say “a surface area formula of the form 3kr[sup]2[/sup]”, of course. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 4:10am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/25 "2013-07-07T04:10:16Z")
> [@Asympotically\_fat](#): > > No it doesn’t really work, consider a regular pentagon inscribed in a square and you’ll quickly see why. But note that we _do_ have that a circle inscribed in a regular pentagon continues this much of the analogy: Perimeter of circle: 2πr; Perimeter of pentagon: 2Fr Area of circle: πr[sup]2[/sup]; Area of pentagon: Fr[sup]2[/sup] Where F = sqrt(5 - 2sqrt(5)). And in general, we have for an N-sided regular pentagon whose sides are at distance r from its center, that its perimeter is 2N \* tan(1/(2N) revolutions) \* r, and its area is N \* tan(1/(2N) revolutions) \* r[sup]2[/sup]. ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 6:48am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/26 "2013-07-07T06:48:16Z")
> [@Indistinguishable](#): > > I like non-linear thought. Yes, but it seems to be lost on you. On the one hand to calculate the area of a circle you can emulate the calculation using a polygon of infinite sides or you can use a constant, pi, that seems to imply a polygon of 3.14159 “sides”. That was the contrast I was after and the sort of thing I wouldn’t expect most people to appreciate. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 7:22am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/27 "2013-07-07T07:22:58Z")
I’m pointing out that the similarity you note shouldn’t be taken as showing a circle to be analogous to a polygon of 3.14159 “sides”. The similarity you note should be taken as showing a circle to be analogous to a polygon of area 3.14159. The 4 in the square’s perimeter and circumference equations does not come from the number of sides of a square; it comes from the area of a square. A polygon of N sides has a perimeter of 2N \* tan(1/(2N) revolutions) \* r, and an area of N \* tan(1/(2N) revolutions) \* r[sup]2[/sup]. π is analogous to N \* tan(1/(2N) revolutions) here, not to N on its own. Plugging 3.14159 in for N would give a perimeter of 9.785r and an area of 4.893r[sup]2[/sup], which doesn’t match a circle at all. Rather, we get the appropriate values as N goes to infinity. It just so happens to be the case that, coincidentally, when N = 4, we also get that N \* tan(1/(2N) revolutions) = 4, as tan(1/(2 \* 4) revolutions) = 1. The proper equation is not “Perimeter = 2 \* number of sides \* r, area = number of sides \* r[sup]2[/sup]”. The proper equation is “Perimeter = 2 \* number of sides \* tan(1 complete revolution/(2 \* number of sides)) \* r, area = number of sides \* tan(1 complete revolution/(2 \* number of sides)) \* r[sup]2[/sup]”. It just so happens that that factor of tan(1 complete revolution/(2 \* number of sides)) happens to equal 1 and thus disappear in the case [and only in the case] when the number of sides is exactly 4. ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 7:31am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/28 "2013-07-07T07:31:40Z")
… ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 7:47am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/29 "2013-07-07T07:47:54Z")
I wasn’t saying it was analogous. I was saying that it was another way of looking at it. I suppose that can be interpreted to mean analogous, but since I’ve already said that a circle clearly doesn’t have sides in any meaningful sense, it should have been obvious that I was trying to imply something that was deliberately logically inconsistent but nevertheless still contained an element of truth. If you don’t see that, that’s fine. But trying to disprove the point is pointless since the “point” never existed to begin with - hence the non-linearity. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 7:56am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/30 "2013-07-07T07:56:01Z")
The element of truth is what I explained in my posts, not that a circle should be looked at as in any sense like a figure with 4 sides. The process of reasoning which caused you to say that is based on looking at formulas involving a square and a 4 in the same place as the corresponding formulas involving a circle and a pi. But these are instances of a general formula involving an N-sided figure and N \* tan(180 degrees/N). The 4 you saw should not be looked at as representing the number of sides of a square; it should be looked at as representing the number of sides of a square _times tan(180 degrees/the number of sides of a square)_. There’s no good reason to think of the 4 in those formulas as representing the number of sides of the square, since in no other case does the role that 4 is playing get filled by a number of sides, _and since we know very well what instead that role is filled by_. If your takeaway was “the role of the 4 in those equations with respect to a square, and accordingly the role of pi with respect to a circle, is to describe the number of sides”, even in a loose sense, you are misguided, and will make wrong statements about every kind of polygon except for a square. It turns out there is a correct explanation of the analogy you noticed, though, where one replaced “number of sides” with a slightly more complicated function of the number of sides. If you dismiss that in favor of the wrong way of looking at it, even after the explanation has been offered, you are doing a foolish, stubborn thing. ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 8:00am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/31 "2013-07-07T08:00:44Z")
The area of a square was offered only for contrast, not to prove anything. You’re the one who seems to have an obsession with proofs. I’m more interested in insights. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 7, 2013, 8:02am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/32 "2013-07-07T08:02:35Z")
And I’m telling you the insight you’re stumbling upon and then, for some reason, past… The insight is that the 4 in “The perimeter of a square of ‘radius’ r is 2 \* 4 \* r, while its area is 4 \* r[sup]2[/sup], and the volume of a square extruded to height h is 4 \* r[sup]2[/sup] \* h” is not meant to describe the number of sides in a square. It’s meant to describe something else, which we understand very well. And therefore you shouldn’t think of a circle as a pi-sided figure, but as a pi-(something else which we understand very well)ed figure. ---
**Author:** ![deltasigma](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@deltasigma](https://boards.straightdope.com/u/deltasigma)\ **Post date:** [July 7, 2013, 8:26am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/33 "2013-07-07T08:26:23Z")
If you want to argue that there is no insight to be gained here that’s fine, but that’s just your humble opinion. If you’re not willing to admit that, that’s your problem, not mine. ---
**Author:** ![Hail\_Ants](https://avatars.discourse-cdn.com/v4/letter/h/dc4da7/32.png) [@Hail\_Ants](https://boards.straightdope.com/u/Hail_Ants)\ **Post date:** [July 7, 2013, 11:45am UTC](https://boards.straightdope.com/t/questions-about-pi/662796/34 "2013-07-07T11:45:55Z")
> [@Dr.Strangelove](#): > > First, it’s easy to exclude lots of fractions as uninteresting, because there are fractions with smaller numbers but with a better approximation. Here’s a list of the first few: > [http://www.isi.edu/~johnh/BLOG/1999/0728\_RATIONAL\_PI/top\_approximations.txt](http://www.isi.edu/~johnh/BLOG/1999/0728_RATIONAL_PI/top_approximations.txt) > > 22/7 and 355/113 are “extra special” in that the next larger numerators with a better approximation are quite a bit larger. For 22/7, the next better fraction is 179/57. For 355/113, it’s 52163/16604 (it jumps over two orders of magnitude!). > > The next “extra special” fraction is 5419351/1725033. The latest episode of Futurama had a great math joke: someplace they had written- **22/7 “Fool’s pi!”** ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [July 7, 2013, 1:54pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/35 "2013-07-07T13:54:04Z")
> [@deltasigma](#): > > Yes, but it seems to be lost on you. On the one hand to calculate the area of a circle you can emulate the calculation using a polygon of infinite sides or you can use a constant, pi, that seems to imply a polygon of 3.14159 “sides”. > > That was the contrast I was after and the sort of thing I wouldn’t expect most people to appreciate. **Indistinguishable** was _cooperating_ with you. He was working entirely within the spirit of what you were attempting. Holy cow, Delta. You do this all the time. You’re one of those people that won’t even let others _agree_ with you peaceably. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [July 7, 2013, 1:55pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/36 "2013-07-07T13:55:50Z")
> [@deltasigma](#): > > If you want to argue that there is no insight to be gained here that’s fine, There it is again! He literally _just told you_ the opposite of this! He said there _is_ an insight you’re on to! ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [July 7, 2013, 3:01pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/37 "2013-07-07T15:01:10Z")
> [@Exapno\_Mapcase](#): > > [How Much Pi Do You Need?](http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/). > > The article talks about the limits of needed precision for pi. The comments on that page give 63 decimals to get to a planck length. This is also approximately how many orders of magnitude of sizes between the Planck Length and the size of the universe. > **[The Scale of the Universe](https://www.newgrounds.com/portal/view/525347)** > > Zoom from the edge of the universe to the quantum foam of spacetime! Are orders of magnitude and decimal places proportional across the board, or are they proportional only around a factor of pi? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [July 7, 2013, 3:22pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/38 "2013-07-07T15:22:39Z")
Frylock, the true insight was lost because ds doesn’t understand that the relationship he was claiming in the end is mathematically wrong. This is no different than all the cranks we get. “Look at my insight!” “Well, that’s interesting in a sense that everybody already understands, but the math is meaningless.” “You fools don’t understand my genius! I’m making a point too subtle for lesser minds!” “Maybe. But greater minds can slice it forty-seven ways and serve it to you for breakfast.” Indistinguishable is explaining to ds something he didn’t see in his marvelous insight because Indistinguishable actually understands the meaning of the math he spouts. Enola Straight, that has to be same. Circumference is pi\*d, where d in this case is the size of the universe. Dividing out pi, which is only about 3, leaves you with a number that is the same order of magnitude. IOW, if there are 63 orders of magnitude between the planck length and the width of the universe, then there will be 63 orders of magnitude needed to have the circumference measured to the same precision. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [July 7, 2013, 3:26pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/39 "2013-07-07T15:26:25Z")
I understand all that, but I believed what ought to be stressed to **deltasigma** was not just “you’re wrong,” but “you’re going in an interesting direction although the way you do it turns out not to work quite right. But still, we can try something like it, and yes, there are interesting things to say here.” That’s what I think **Indistinguishable** was doing, and **deltasigma** ought to have taken this as a positive development. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 7, 2013, 4:34pm UTC](https://boards.straightdope.com/t/questions-about-pi/662796/40 "2013-07-07T16:34:01Z")
Orders of magnitude are really just a count of the number of digits, in any context. Well, to within some quibbles that come up from discretization: 9 and 10 are roughly the same order of magnitude, and 9 and 1 are roughly an order of magnitude apart, since number of digits is an integer but orders of magnitude needn’t be. [Previous page](https://boards.straightdope.com/t/questions-about-pi/662796.md?page=1) [Next page](https://boards.straightdope.com/t/questions-about-pi/662796.md?page=3)