Geometrical/Combinatorial interpretation of $ \pi = \frac{27}{4 \sqrt 3} \sum_2^∞ \frac{1}{C(k)} $

Question

I found this question on here, but no real answer was given. Is there a geometrical or combinatorial interpretation of this result?

$ \pi = \frac{27}{4 \sqrt 3} \sum_2^∞ \frac{1}{C(k)} $

Where $ C(k) $ are the Catalan numbers. This result can be derived from here and this is the oeis reference