Where’s the Calculus?Īrchimedes wasn’t “doing calculus” but he laid the groundwork for its development: start with a crude model (square mimicking a circle) and refine it. There’s even better formulas out there too.
Some people use 22/7 for pi, but now you can chuckle “Good grief, 22/7 is merely the upper bound found by Archimedes 2000 years ago!” while adjusting your monocle. If you enjoy fractions, the mysteriously symmetrical 355/113 is an extremely accurate (99.99999%) estimate of pi and was the best humanity had for nearly a millennium. The midpoint puts pi at 3.14185, which is over 99.9% accurate. His final estimate for pi, using a shape with 96 sides, was: He began with hexagons (6 sides) and continued 12, 24, 48, 96 until he’d had enough (ever try to take a square root using fractions alone?). So Archimedes had to slave away with these formulas using fractions. Unfortunately, decimals hadn’t been invented in 250 BC, let alone spreadsheets. And after 17 steps, or half a million sides, our guess for pi reaches Excel’s accuracy limit. After 7 steps (512 sides) we have the lauded “five nines”. Let’s assume pi is halfway between the inside and outside boundaries.Īfter 3 steps (32 sides) we already have 99.9% accuracy.
Weird ways to calculate pi download#
Starting with 4 sides (a square), we make our way to a better pi ( download the spreadsheet):Įvery round, we double the sides (4, 8, 16, 32, 64) and shrink the range where pi could be hiding. Using the Pythagorean theorem, side 2 + side 2 = 1, therefore side = $\sqrt$ and 1, we can repeatedly apply this formula to increase the number of sides and get a better guess for pi.īy the way, those special means show up in strange places, don’t they? I don’t have a nice intuitive grasp of the trig identities involved, so we’ll save that battle for another day.
We don’t know a circle’s circumference, but for kicks let’s draw it between two squares: (He actually used hexagons, but squares are easier to work with and draw, so let’s go with that, ok?). But he didn’t fret, and started with what he did know: the perimeter of a square. What’s behind door #3? Math! How did Archimedes do it?Īrchimedes didn’t know the circumference of a circle. Draw a circle with a steady hand, wrap it with string, and measure with your finest ruler.Pi is the circumference of a circle with diameter 1. I wish I learned his discovery of pi in school - it helps us understand what makes calculus tick. Could you find pi?Īrchimedes found pi to 99.9% accuracy 2000 years ago - without decimal points or even the number zero! Even better, he devised techniques that became the foundations of calculus. But what if you had no textbooks, no computers, and no calculus (egads!) - just your brain and a piece of paper.
Sure, you “know” it’s about 3.14159 because you read it in some book.
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