General Knowledge
Why is pi the same number for every circle?
- All circles are the same shape scaled
- It was defined that way
- Coincidence
The answer is All circles are the same shape scaled. Every circle is a scaled copy of every other, so the ratio never changes.
Because all circles are the same shape. A big circle is a scaled-up copy of a small one, and scaling multiplies every length by the same factor - so the circumference and the diameter both grow in exactly the same proportion and their ratio never moves. That is why pi is a property of circularity rather than of any particular circle.
It is irrational, proved in 1761, meaning it cannot be written as a fraction and its decimal expansion never repeats. It is also transcendental, proved in 1882, which is the stronger statement that it is not the solution of any polynomial equation with whole-number coefficients - and that result finally killed the ancient problem of squaring the circle, showing it is not merely difficult but impossible with compass and straightedge.
Attempts to legislate it are the entertaining footnote. In 1897 the Indiana House of Representatives unanimously passed a bill based on an amateur's claimed squaring of the circle, which implied a value of pi around 3.2. A mathematics professor happened to be at the statehouse on other business, read it, and briefed the senators, who postponed it indefinitely. It has never been repealed - merely never taken up again.