I am not privy to how Mathematica does things under the hood, but it is indeed true that one does not need
$\pi$ to draw a circle. On the conceptually simpler end of things, there is e.g. Minsky's method, whose (naive) Mathematica implementation might go like
With[{h = N[1/16]},
Graphics[Line[NestList[{{1, -h}, {h, 1 - h^2}}.# &, {1, 0}, Round[8/h]]]]]
and on the fancier end of things, one can use non-uniform rational B-splines (NURBS), which is mathematically equivalent to parametrizing a circle with the stereographic projection:
With[{n = 150},
ParametricPlot[{1 - u^2, 2 u}/(1 + u^2), {u, -n, n}, PlotRange -> All]]
Jim Blinn gives a good survey of various methods here. All of this is independent of computing
$\pi$.