A droplet of liquid placed on a flat high-energy solid surface spreads to give a thin film so that no macroscopic droplet shape exists. On a chemically identical solid surface with only the geometry changed to a cylinder, the same droplet can have an equilibrium conformation. When the equilibrium conformation is of a barrel type, the profile of the droplet changes rapidly in curvature as the three-phase contact line is approached and the direct measurement of the contact angle is difficult. This work considers the theoretical profile for barrel-type droplets on cylinders and discusses how the inflection angle in the profile depends on droplet parameters. Experimental results are reported for poly(dimethylsiloxane) oils on a range of fiber surfaces and these are used to estimate the equilibrium contact angle from the inflection angle. The drop radius and volume dependence of the inflection angle is confirmed.