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I am absolutely confused about trying to calculate circumference. And I do not mean using the math formula, I mean back in old days when people had very primitive tools, and had to make the discoveries.

In order to create a circle you can take a long strip of paper and try to fold it into a circle. By knowing paper's length you can know circumference. However how would you go about putting the piece of paper into perfect circle to measure the diameter? No matter what you do you might be just a millimeter off while trying to measure the diameter.

Now let's say you take a compass and draw a circle. You will be able to easily measure the diameter, however you will not be able to fold a strip of paper into a perfect circle, again it might be just a millimeter off.

Third way I can think of they used is to first draw a square inside and outside of the circle you draw with compass. Then you changes it into pentagon, then hexagon and so forth and use math to try to find the circumference. But again I think it does not allow you to calculate with perfect precision (no calculators, primitive age).

Due to that no matter what you do you will not be able to perfectly derive formulas to calculate circumference if given diameter or vice versa. So how exactly did the people derive these formulas and make perfect circles with all the values know and discover pi?

Sorry I tried searching but didn't find anything. I would really love to know the answer.

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  • $\begingroup$ I am also fascinated by Pi. This page is quite good ualr.edu/lasmoller/pi.html $\endgroup$ – Claude Leibovici Nov 2 '13 at 15:19
  • $\begingroup$ There are of course various levels of knowledge that have been gained: (1) The ratio of perimeter to diameter is constant (let's call it $\pi$). (2) $\pi> 3$ (3) $\pi\approx\frac{22}7$ or other approximations. (4) Finding a method to systematically find better approximations (5) The ratio of area to square of the radius is also the same constant $\pi$ (6) The same constant $\pi$ plays a role for sphere surface and volume (7) $\pi$ is irrational (8) squaring the circle is impossible / $\pi$ is transcendental (9) $\pi\approx 3.1415926535897932384626433$. - Each of these is a historical step. $\endgroup$ – Hagen von Eitzen Nov 2 '13 at 16:06
  • $\begingroup$ I went to a talk about how the ancient Chinese did this by precisely the method you described - inscribed and circumscribed $n$-gons, where $n$ was very large. Also, it is not totally obvious the the length of the circumscribed polygon is longer than the circumference of the circle. $\endgroup$ – Stephen Montgomery-Smith Nov 2 '13 at 20:52
  • $\begingroup$ We get modern precision with calculus and series.mathworld.wolfram.com/PiFormulas.html before calculus, I imagine values were very rough and I think the estimate 377/120 might have been as good as it got. (Not entirely sure how they got that.) $\endgroup$ – fleablood Feb 19 '16 at 4:39
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As to your assumption that they would never be able to shape it perfectly into a circle, suppose you had a circular block of wood. You cut a piece of paper slightly longer than the circumference, wrapped it around the wood, and drew a line on the paper using the circle as a guideline. Now you measure the length of the line. This would be trickier than it sounds, but it would be possible with the material they had, and would be very accurate.

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  • $\begingroup$ Wow I had not though that you can make a compass like tool to actually make a circular block of wood, instead I kept thinking that they would try using compass to draw a circle and then try to align the piece of paper on it. However I think your solution should work. I will try doing so over the weekend to see if it is feasible and/or accurate. $\endgroup$ – Quillion Nov 6 '13 at 18:46
  • $\begingroup$ Try using a lathe to turn the wood. You can also turn metal, if you have access to a metal-turning lathe. $\endgroup$ – Hosch250 Nov 6 '13 at 19:48
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The question asked is related, I guess, to that of the Portolan Charts and how they could have been drawn so accurately. It was not just a matter in those days of just 'taking' a compass rose. They had to draw one. And there was no square paper or spreadsheet programs. My answer would be that it is because of the amazing regularity of the 'behavior'of numbers that they found their way.

I have watched the building of a traditional house in an African country, which started out by fixing a pole in the ground. To that pole was attached another pole, with a ring at one end, which fit just around the central pole, while the other end was used to measure the distance of the (mud) bricks to the central pole. After one layer of bricks had been laid, the pole was placed on a support of one brick high, etc., which shows the use of the radius to the circumference.

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