When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
T = 2π√(l/g)
T/2π = √(l/g)
(T/2π)^2=l/g
g = l/(T/2π)^2
g = l/(T^2/4π^2) = 4π^2xl/T^2
Now substitue T with 2 and l with 1 an you get
g = 4π^2x1/2^2 = π^2
It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated.
There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics.
If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ?
I'll try to see these units you linked thanks
g in imperial units is 32 after all. g has units; pi does not
The pendulum is a device that relates pi to gravity.
PI = sqrt(g/L)
g = 9.81. L=1
or
g = 32.174. L=3.174
Either way works to approximately pi. There is a particular length where it works out exactly to pi which is about 3.2 feet, or about 1 meter. My point was that equations like that remain true regardless of units.
The reason pi squared is approximately g is that the L required for a pendulum of 2 seconds period is approximately 1 meter.