How does the square-root of the constant G equal Pi?

It doesn't.

g, the acceleration due to gravity is 9.81m/s². The square root of this is 3.132 approximately. Pi is 3.142 approximately. That's a 0.3% difference. It's a complete coincidence that they are close.

Of course, the value of g depends completely on what units you are using. g = 33 ft/s² approx, which is nothing to do with Pi.

I wondered about that, but that's an acceleration not a constant. G is the gravitational constant, and is something like 6.6742 x 10 to the minus 11 Nm squared kg to the minus 2. https://en.wikipedia.org/wiki/Gravitational_constant This is a constant regardless of the units, but its square root has no connection I can see with pi. Over to you now Peasy. Your turn.....