Useful length of Pi?

Question

Not sure where this really fits, so am trying Mathematics first. Feel free to migrate to another StackExchange forum if more appropriate elsewhere. So I was listening to a podcast yesterday that was discussing the progression of calculating Pi from original rough approximations of 3.x, to the current over 100 trillion (or maybe more). I can comprehend the desire to calculate Pi to as far as possible, but what is the practical limit of the number of digits in Pi's usage in science, engineering, chemistry, physics, etc.?

Answer 1

I read once that you can get decently accurate results in astronomy just by using $\pi \approx 3$.

You can get a feel for how accurate your approximation to pi needs to be with some calculus.

Area of circle is $A=\pi r^2$, so $dA=d(\pi) r^2$, so $\frac{dA}{A}=\frac{d\pi}{\pi}$.The relative error in the area is the relative error in pi. A 10% error in one will give you a 10% error in the other.

The force between two electrons is $F=\frac{e^2}{4\pi \epsilon_0}$ so $\frac{dF}{F}=\frac{4\pi\epsilon_0r^2}{e^2}\frac{e^2}{4\epsilon_0r^2}\frac{-d\pi}{\pi^2}=\frac{-d\pi}{\pi}$

$\zeta(2)=\frac{\pi^2}{6}. \frac{d\zeta(2)}{\zeta(2)}=\frac{2\pi d\pi}{6}\frac{6}{\pi^2}=\frac{1}{3}\frac{d\pi}{\pi} $

A 10% error in $\pi$ is on the order of $0.3$ so a value between $2.85$ and $3.42$ should be fairly accurate.