How do you determine how many digits of pi are necessary? It is said that you only need to calculate pi to 62 decimal places, in order to calculate the circumference of the observable universe, from its diameter, to within one Planck length.

Most of us are probably dealing with more terrestrial things. Automobile tires, for example. And we can usually tolerate more sloppy calculations. Say, to within one micrometer.

Given a specified scale and/or desired precision, how can we figure out how many digits of pi we really need?
Using $\Delta$ for errors and usual notation otherwise for the circumference of a circle,

$$\Delta C = 2 R \Delta\pi + 2\pi\Delta R$$

If you want $\Delta C$ within some error margin, you usually have to worry first about minimizing measurement error for the radius $\Delta R$. But if that is no longer a constraint, then worry about $\Delta \pi$. 

For example, suppose your target error $\Delta C = 1 \ \mu m$ but you can only measure $R$ within one $\mu m$. Then your error for $C$ is $2\pi \ \mu m$ before we even consider $\Delta \pi$. If you could measure $R$ within say $1/10 \ \mu m$, then we can begin to worry about solving for the required number the digits for $\Delta\pi$:

$$\text{ Target value of } \Delta \pi \leq \frac{\Delta C - 2\pi\Delta R}{2R}$$
a quantity that is now positive and meaningful. Suppose in our example that $R = 1 \ m$, then $\Delta \pi \leq 1.86 \times 10^{-7}$ and thus we would want $\pi$ accurate to at least $7$ decimal places.
Um, that depends on the amount of accuracy you need.  That's usually expressed as "significant digits".  If you need accuracy within .01% = 0.0001 margin of error, that is "four significant digits".  Everything, not just pi, can be estimated to four significant digits so it suffices to just use the first four digits of pi.  If you need higher accuracy you use more digits.