You might remember pi from your high school maths lessons: it’s something to do with circles, right?

In honour of Pi Approximation Day, we’re going to refresh our memories and even learn something new about the secrets of pi.

## What is pi?

Imagine a circle – or grab a plate or a piece of paper and a pencil, and even a compass if you’re fancy, and draw yourself one. The circle’s circumference is the length of the line that runs around the outside of the circle.

The diameter is the length of a line that runs from one side of the circle to the other, through the circle’s centre. Half of the diameter, or the distance from the centre of the circle to its edge, is called the radius.

Pi can be defined as the ratio of a circle’s circumference to its diameter, or – to put it another way, circumference/diameter = pi.

The letter pi was probably chosen to represent this number because it relates to the concept of the ‘perimeter’, which is the distance around the edge of a shape – the circumference is actually the perimeter of a circle – or ‘periphery’, which also refers to an external boundary. These words both begin with P in English, and they happen to begin with pi in Greek.

But how much actually *is* pi? Well, it’s *approximately* 3.14.

## What’s special about pi?

So why don’t we just say pi = 3.14?

Well, pi *isn’t* actually 3.14, or 3.14159, or even 3.14159265358979323846 (that’s pi to the first 20 decimal places). Pi is an irrational number: its decimal representation never ends and never repeats.

Pi has been calculated to trillions of digits, but that’s still not *exactly* pi. It’s just a very, *very* good approximation.

Another rather handy approximation of pi is a simple fraction: 22/7.

And *that’s* where Pi Approximation Day comes from – 22/7 looks like the 22^{nd} of July in a day-month date format.

Plenty of people also celebrate Pi Day on 14March, because 3.14 looks like March 14^{th} in month-day format … but we’re Australian and, anyway, Australian winter is a more appropriate time to eat pie.

## Why does pi show up in so many places?

So, pi is a number that describes the relationship between a circle’s diameter and its circumference – so far so good.

Thus, unsurprisingly, pi is in lots of geometry formulas about the area and volume of circles, spheres and other round-ish shapes.

But pi also appears in all sorts of other places that seemingly don’t have much to do with circles. In probability, the square root of two pi is part of the formula for normal distribution, aka the bell curve. Coulomb’s constant, which describes the force between two electrical charges, is defined as one divided by four pi times epsilon zero (a measure of how much electrical charge the empty space can store). Pi also appears in the formula for the cosmological constant, used by astrophysicists to account for the theorised dark energy that may be driving the expansion of the universe.

And there’s more – pi shows up in the maths underlying epidemiology, fluid dynamics, and even quantum mechanics.

## Is pi just magically at the centre of everything?

Many concepts involving pi involve circles or spheres. A physical force or an electrical field spreading out evenly in all directions is actually an invisible sphere.

But pi is also related to all kinds of oscillations and cycles – systems and patterns where things move back and forth within a range of values. Examples include the way a coiled spring extends and contracts, or how a pendulum on a grandfather clock swings from side to side.

Pi is central to describing the sine wave, an undulating curve that gracefully oscillates between one and negative one. And lots of phenomena in science have a sine-shaped wave somewhere in them, from electrical voltage to sound waves to seasonal disease cycles to the electromagnetic spectrum, which includes visible light, x-rays, microwaves and more.

Happy Pi Approximation day – and we hope you’ll join us in taking the chance to have a circular, pastry-based snack, and marvel at this impressive number.

Originally published by Cosmos as Pi Approximation Day – what’s behind the magic? (Hint; maths not apples)

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