Specifically, what is Pi besides a number allowing you to calculate the circumference of a circle?

Question

What is Pi? Why is Pi the only number able to calculate the circumference of a circle? Also, how was Pi discovered?

Answer 1

Usually, Pi ($\pi$) is defined to the number that is the ratio of the circumference of a circle to its diameters. That is, if you measure a circle perfectly and find the circumference equals $C$ and the diameter equals $d$, then you will have $$\pi = \frac{C}{d}.$$

Now, there are many other good numbers out there for determining the circumference of a circle. For example, the number $\tau$ is the ratio of the circumference of a circle, $C$, to its radius, $r$. That is $$\tau = \frac{C}{r}.$$ However, since the diameter is twice the radius ($d = 2r$) this all comes down to the same idea and we have $$\pi = \frac{C}{d} = \frac{C}{2r}.$$ Therefore we have $\tau = 2\pi$.

The value of $\pi$ (or at least approximations) were most likely discovered through measuring the circumference and diameter very precisely. As these measurements were carried out, someone probably noticed a pattern and that the ratios were very close. He or she then began to measure more precisely and noticed that the ratio was in fact constant.