A visual introduction
π (pi) is the ratio of a circle's circumference to its diameter — the same number, no matter how big or small the circle. Below are four different ways people have calculated it, from ancient geometry to modern random sampling. Drag the sliders and press the buttons to see each one work.
01 · Definition
Wrap a circle's circumference around a straight line made of copies of its own diameter. It takes exactly 3 full diameters, plus a little more — that "little more" is 0.14159…, and it never changes, for any circle.
02 · Geometric method, circa 250 BCE
Archimedes squeezed π between two numbers: the perimeter of a polygon drawn just inside a circle (too short), and one drawn just outside it (too long). Add more sides, and both close in on the true value.
03 · Infinite series, 1670s
π can also be built from an endless sum of shrinking fractions. Each term nudges the running estimate above and below the true value, settling in slowly as more terms are added.
04 · Statistical method, 20th century
Throw random darts at a square that has a circle inscribed in it. The fraction that land inside the circle approximates the circle's share of the square's area — which is exactly π ⁄ 4.