Method 1 · ~250 BC
Squeezing π with polygons
Archimedes had no decimals, no calculators, and no algebra in the modern sense. He still pinned π down to two correct digits — using nothing but shapes.
Hexagon — where Archimedes began
Each side of the inscribed hexagon equals the radius, so its half-perimeter is exactly 3. A surprisingly clean starting point — and the reason he chose it.
The big idea
Imagine a circle. You want to know how far it is around the outside. That distance — the circumference — equals π × the diameter. So if the diameter is 1, the circumference is exactly π. Just measure the circumference and you've got it!
Easier said than done. A circle is curved. You can't lay a ruler against it. So Archimedes did something clever: he replaced the curve with something he could measure — straight lines.
Two polygons, one circle
He drew a regular polygon inside the circle, with each corner just touching it (the inscribed polygon). Then he drew another polygon outside, each edge just kissing the circle (the circumscribed polygon).
The circle's circumference has to be bigger than the inside polygon's perimeter (which cuts corners) and smaller than the outside polygon's perimeter (which juts out). So π is squeezed between two numbers you can actually compute.
Add more sides
A triangle is a lousy circle. A hexagon is a little better. With 96 sides you can barely tell the polygons from the circle at all — and that's exactly where Archimedes stopped. Try sliding the dial up there above. Watch the bracket close.
His final answer: π is bigger than 3 + 10⁄71 and smaller than 3 + 1⁄7. In modern decimals, that's between 3.1408 and 3.1429. He nailed two decimal places of an infinite number, by hand, with shapes.
The math, if you want it
For a regular polygon with n sides drawn inside a circle of radius 1, the perimeter is 2n · sin(π/n). Drawn outside, it's 2n · tan(π/n). Half of each gives an estimate of π. As n grows, both estimates converge on π — sin and tan get closer and closer to their angle in radians for small angles.
Of course, Archimedes didn't have sin or tan. He used a recursive doubling trick that started with a hexagon (n = 6) and doubled to 12, 24, 48, 96 — computing each new perimeter from the previous one with just square roots. Working with square roots, by hand, in Greek numerals. Astonishing.
Wait — did we just invent calculus?
Here's a thought I floated to my running group on a long, slow trail:
What is a circle? A circle is the locus of points equidistant from another point in a single plane. And since you can draw a line between any two points, you can think of the circumference of a circle as the sum of the lengths of the infinite-sided polygon that makes up the circle.
One of the engineers in the group shot back: "Guys, BHB just invented calculus."
He was teasing — but also kind of right. The idea that a smooth curve is the limit of straight pieces, and that you can find its true length by adding up infinitely many tiny segments, is the seed of calculus. It's how we compute the length of any curve, the area under any wavy line, the volume of any oddly-shaped solid. Archimedes was doing this 1,900 years before Newton and Leibniz formalized it into the notation we use today.
And the same thread runs through trigonometry: look back at the formulas above — the polygon perimeters involve sin and tan. Those aren't a separate subject. They're the math of polygons described with angles. So in this one little exercise, geometry, trigonometry, and the seed of calculus are all the same idea looking at the same circle from different sides.
Pretty good for shapes.
Try this
- Hit Animate and watch the polygons collapse onto the circle.
- Find the smallest n where the first three digits (3.14) match exactly.
- Turn off the inscribed polygon. Notice how the outside one shrinks toward the circle from above.