Why π is what it is

2026-05-01

Wrap a string around a tin can. Unspool it. Compare it with the diameter of the can. The string is always a little more than three times longer. Try a smaller can, or a planet. The same number.

The number line under the circle is marked in diameters. The 1 tick is one diameter from the start, the 2 tick is two, and so on. The π tick - the one in red - is at three diameters and a bit. After one full revolution, a circle rolling without slipping has covered exactly its own circumference, which is exactly π diameters. So it lands on the π tick. Always.

Drag the size

Move the d slider to make the circle bigger or smaller. The whole figure rescales: the ticks spread or shrink, the circumference grows or shrinks, the trace grows or shrinks. The one thing that doesn't change is the alignment - the rolled trace ends precisely at the π tick, every time. The ratio stays put because π is the ratio. It is a property of the shape, not of the size.

Why ratios are special

Most numbers in the world depend on size. The area of a tin lid depends on the lid. The weight of a watermelon depends on the watermelon. But ratios - dividing one length by another - cancel out size. If you double everything in a figure, every length doubles, so every length-divided-by-length stays the same. Ratios are the part of a shape that survives scaling.

Every circle in the universe is the same shape, just scaled. So every circle has the same circumference-to-diameter ratio. There is exactly one such number, and we call it π.

What number is it?

π begins 3.14159 26535 89793 ... and never stops and never repeats. You cannot write it down as a fraction of two whole numbers. You cannot write it down at all, exactly. The Babylonians used 3. The Egyptians used 256/81 ≈ 3.16. Archimedes squeezed it between 223/71 and 22/7 by inscribing polygons inside and outside a circle. Today we know trillions of digits and none of them have ever shown a pattern.

None of that matters here. What matters is that there is such a number, that it is the same for every circle, and that you can see it land on the tick.

C = π · d. Whatever the circle. Whatever the size.