Gjergji Zaimi's note on the average value of the Euler phi-function reminds me of some other appearances in Number Theory that don't have any obvious circles in them. The probability that two randomly selected positive integers are relatively prime is $6/\pi^2$ (where "randomly selected positive integers" is shorthand for, fix $n$, choose two integers uniformly and independently from $[1,n]$, then take a limit as $n$ goes to infinity). And the density of the squarefree integers (integers divisible by no square number other than 1) is also $6/\pi^2$.

In a horse race, a trifecta is a choice of horses finishing first, second, and third, in order. If the horses are numbered from 1 to $n$, one can ask for the number $S(n)$ of trifectas in geometric progression (e.g., 4-6-9 is a trifecta in geometric progression when interpreted as horse number 4 finishing 1st, horse number 6 finishing 2nd, etc.). I proved that $S(n)=(6/\pi^2)n\log n+O(n)$, and I attributed the appearance of $\pi$ in the formula to the circular portion of the race track.