# How exactly do you measure circumference or diameter?

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.

• I am also fascinated by Pi. This page is quite good ualr.edu/lasmoller/pi.html – Claude Leibovici Nov 2 '13 at 15:19
• 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. – Hagen von Eitzen Nov 2 '13 at 16:06
• 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. – Stephen Montgomery-Smith Nov 2 '13 at 20:52
• 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.) – fleablood Feb 19 '16 at 4:39