You know the name and you know at least some of the number. Pi, a never-ending value a little more than three, is the ratio of a circle’s circumference to its diameter. It’s also irrational, meaning that pi — denoted with the Greek symbol π since the mid 1700s — cannot be expressed as a common fraction like 1/2 or 3/5. The number never ends, and in the intervening centuries we’ve figured out a few trillion digits behind the 3.
But what is π? Maybe the best way to understand this crucial mathematical constant is to see where it comes from.
If π is the ratio between a circle’s circumference — the distance around the circle — and its diameter, then seeing both laid out should show us what π is. Make a circle’s diameter 1 and then lay out the circumference. How many diameters is that? A little over 3, or 3.14159…!
Another way to think of π is to in radians, the standard unit of angular measurement. 1 rad is equal to the angle that an arc the same length as the circle’s radius makes. A circle’s radius is one-half its diameter, so you should be able to fit π radians into half a circle. The math works out — you can fit three radians and some change (0.14159…to be precise) in a half-circle!
Another way to think of the ratio of a circle’s circumference and diameter is to imagine the classic “sine wave.” You’ve probably seen these waves blinking and flowing on sci-fi gadgets or in your trigonometry class. A sine wave is the 2D graph of the sine function, which describes the ratio of the length of a side that is opposite a right triangle’s angle to the length of the longest side of that triangle.
The sine wave can be traced out by tracking the coordinates of a point running along a circle with a radius of one. And like the GIF above, when half the circle has been traced, exactly π distance has been covered! See? GIFs are always a bit easier to digest than high school trig.
Remember, it’s OK to be irrational sometimes.