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

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##