Pi

Pi, or π, is the ratio between the circumference and the diameter of a circle. π is not a number, because the numerical value of π depends on context: which circle is under consideration, and how precise are one's measurements? π should be thought of as more like a number-valued variable, which stands only for a very specific, limited, class of things.

The traditional concept

Standard mathematics says that π is the ratio between the circumference and the diameter of a perfect circle.

Standard mathematics "measures" this ratio using the methods of calculus.

Standard mathematics' notion of π is an irrational number---a number which cannot be written as a ratio of two whole numbers.

Thus in standard mathematics,

π = 3.141592653589793238462643383279502884197169399375105 8209749445923078164062862089986280348253421170679...

where the "..." signifies that the digits go on forever.

Examples

π shows up in many places, besides just circles.

a disk of radius $r$ has area $\pi r^{2}$

a cone with height $h$ and with base of radius $r$, has area ${\frac {1}{3}}\pi r^{2}h$

a sphere of radius $r$ has volume ${\frac {4}{3}}\pi r^{3}$, and surface area $4\pi r^{2}$

the area under a Gaussian distribution is proportional to some power of $\pi ^{1/2}$.