How is the formula for the circumference of a circle derived?

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Answer 1 · 2016-01-24T19:15:36

Essentially by the definition of $pi$

Any circle is similar to any other circle, so the ratio between the length of the circumference of a circle and its diameter will always be the same.

This ratio is the number we call $pi$ ("pi").

Then since the diameter is twice the radius we get the formula:

$c = 2pi r$

expressing the length of the circumference in terms of the radius.

$pi$ is an irrational number. That is it is not expressible as a fraction $p/q$ for any integers $p$ and $q$ .

There are some useful approximations, especially the well known $22/7 = 3.bar(142857)$ and the lesser known but much better $355/113 ~~ 3.1415929$ .

To a few more decimal places we find:

$pi ~~ 3.141592653589793238462643383279502884197$