Question

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

Answer 1

Essentially by the definition of `pi`

Explanation:

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`