Pi

One ratio, the same for every circle.

Pi is the ratio of a circle's circumference to its diameter. The ratio is identical for every circle, which is what makes it a constant rather than a measurement.

Irrational and transcendental

Pi is irrational: it cannot be written as a ratio of two integers, proved by Lambert in 1761. It is also transcendental, proved by Lindemann in 1882 — it is not a root of any polynomial with rational coefficients. That second result settled squaring the circle as impossible with compass and straightedge.

Estimating pi

Archimedes trapped the circle between inscribed and circumscribed polygons, narrowing the value as sides were added, and reached 223/71 < pi < 22/7. The method is a good classroom demonstration because the bounds tighten visibly as the polygon gains sides.