What is 2*pi in the pendulum time formula?

Answer

That factor of 2.pi comes up regularly when you talk about the concept of frequency. The reason is very simple:
When we think of a frequency in daily life, we interpret it as the number of times something happens per second. We then express that frequency f in Hertz (or kHz, MHz…)
If you describe a periodic phenomenon (such as a pendulum) mathematically, however, you automatically arrive at a different kind of frequency from physics, the angular frequency ω. It is expressed in radians per second. For example, a frequency always appears in mathematical derivations, whether it concerns mechanical vibrations or alternating voltages.

Because 2pi radians fit in one circle, the relationship between those two frequencies is:

ω = 2 pi f

so in summary: those two pi come from the fact that physics likes to work with the angular frequency ω in radians per second, and humans like to work with the frequency f in hertz. It’s just the proportionality factor that converts one thing into another.