How did someone come up with the number “Pi”?

Question

How did someone come up with the number “Pi”?

In: Mathematics

Answer 1

You have a circle and its diameter is one unit. You want to measure the distance around the circle using geometry.

You can’t measure the distance around the circle, but you *can* measure the length of a straight line. So you draw a shape whose sides are straight lines — a regular polygon, with all the sides and angles equal.

In fact you draw two polygons, inside and outside the circle, [like this](https://en.wikipedia.org/wiki/Method_of_exhaustion#/media/File:Archimedes_pi.svg).

Now there are three important facts about drawing polygons like this:

– You know the distance around the outside polygon will be longer than the distance around the circle.
– You know the distance around the inside polygon will be shorter than the distance around the circle.
– You know enough about the lengths and angles involved to figure out the length of the polygons’ sides with some formulas.

Then you figure out another formula for a polygon with twice as many sides, which you can apply repeatedly. As the polygons have more sides, they start to look like circles; the distance around them gets closer and closer to the actual distance around the circle.

Archimedes used this technique about 2200 years ago, and figured out that pi is somewhere between 3+10/71 and 3+10/70.

That’s the basic idea of one way you could calculate pi to as much precision as you need. Of course our calculating ability has improved vastly since Archimedes (who didn’t even have the Arabic decimal arithmetic system that we use today). And we’ve also discovered new techniques that produce more precision with less arithmetic.