Definition:Pi/Definition 1

Definition
Take a circle in a plane whose circumference is $C$ and whose radius is $r$.

Then $\pi$ can be defined as $\pi = \dfrac C {2r}$.

(It can be argued that $\pi = \dfrac C d$, where $d$ is the circle's diameter, is a simpler and more straightforward definition. However, the radius is, in general, far more immediately "useful" than the diameter, hence the above more usual definition in terms of circumference and radius.)

Uniqueness of Pi
Note that $\pi$ is defined on a per-circle basis. For each circle with its own circumference $C$ and diameter $d$, $\pi$ is defined as the ratio between the two. It is conceivable, then, that $\pi$ has a different value for each circle. It is also true, however, that All Circles are Similar and thus proportional in size. Thus, the value of $\pi$ is consistent between any two circles, and the constancy of $\pi$ is proven.

Also see

 * Equivalence of Definitions of Pi