I understand the mathematical definition of Pi, but why does it end up being used in so many formulas and applications in math, engineering, physics, etc? What does it unlock?

Edit: I understand Pi is the ratio of circumference to diameter. But why is that fact make it important and useful. For example it shows up in the equation for standard normal distribution. What does Pi have to do with a normal distribution. That’s just one example.

In: 3

It’s just the ratio of a circle’s diameter to its circumference.

This comes up a lot in various disciplines because so many things move in circles, oscillate around a central value in a circular manner, or collapse into circle/spheres as the simplest shape.

Your classic sinusoidal wave is just moving around a circle that’s also moving, and so pi pops up in all manner of wave behaviors and vibrations from the smallest electron to the chaotic magnetic fields of giant stars.

Pi is ratio for diameter to it radius. And it unlocks just that – understanding how to evaluate that. You would see PI everywhere where it is nessessary to work with circles or their parts. Interesting part comes from understanding that many things can be painted inside circle and thus can use some of circle related equasions. As enginiring and physics havely using trigonometry it is not surprise tha PI appears in many many formulas.

Pi is important be cause it is a fundamental attribute of nature and very useful. Without PI it is impossible to comprehend or to calculate many simple things, like how a circuit works or electricity is generated or how elementary particles interact with each other, or even how to accurately design a wheel.

So, Pi is, by definition, related to circles and therefore to spheres as well.

So anything circular, like planetary orbits, is also related to Pi.

And things that are cyclic, like [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion).

And a sphere is the collection of all points that are a certain distance from a central point. So anything that propagates equally in all directions (such as gravitational and electric fields) also deals with spheres, and therefore relate to Pi.

There are many more such examples.

I’d say it’s just one of the “magic numbers” of the universe like the speed of light and the golden ratio. ‘They just are’.

Edit: And just in case I missed the “ELI5” part: pi multiplied by diameter of a circle always provides the circumference of the circle.