It might be famous that $\pi^2$ is a good approximation to the gravitational acceleration in the unit "meter per second squared".
My explanation for this is the seconds pendulum, which was proposed as the definition of "meter" back to the birth of the unit, competing with the meridional definition.
However, I don't see why the two competing definitions of meter have to be close to each other. I don't know any pre-existing unit of length for their reference. And in the meridional definition, the "ten-millonth" seems to be chosen just for convenience, doesn't it?
So why the "meter" according to the seconds pendulum and the "meter" according to the meridian turn out to be close? Is it purely a coincidence?