ctr = {(Exp[Pi*0.5] + 1)/2, 0};
radius = (Exp[Pi*0.5] - 1)/2;
pp = PolarPlot[
  Evaluate@Table[Exp[(t + 2*Pi*i/120)*0.5], {i, 1, 120}], {t, -Pi,
   Pi}, RegionFunction -> (Norm[{#, #2} - ctr] <= radius &)]
Total[ArcLength /@ Cases[pp, _Line, All]]

ctr = {(Exp[Pi*0.5] - 1)/2, 0};
radius = (Exp[Pi*0.5] - 1)/2;
pp2 = ParametricPlot[
  Evaluate[Table[{t, t (Tan[2 j Pi / 120])}, {j, 1, 60}]], {t, -5, 5},
   RegionFunction -> (Norm[{#, #2} - ctr] <= radius &),
  PlotRange -> All]
Total[ArcLength /@ Cases[pp2, _Line, All]]