The measurement of the equilibrium contact angle of a small droplet of fluid partially wetting a flat solid surface provides information on the solid–liquid interfacial energy. However, if the spreading power,S= γSV− (γSL+ γLV), of the surface is positive the liquid spreads completely, no equilibrium contact angle exists, and the resulting thin film has an ultimate thickness determined by Van der Waal's forces. On a chemically identical solid surface with only the geometry changed to a cylinder the same droplet of fluid which completely wets the flat surface can provide an equilibrium conformation. The indefinite spreading tendency is inhibited and the equilibrium is not necessarily a thin sheathing film about the fiber, but can have a macroscopic profile. On a high energy cylindrical surface a barrelling type droplet is only approximately spherical in cross section. Near the three phase contact line the curvature can change sign and measurement of the contact angle becomes difficult. In this work we consider the theoretical profile for such droplets and calculate the extent to which decreasing the fiber radius changes the surface energy and the maximum slope of the profile. We suggest that measurements of the inflection angle in addition to the reduced thickness and reduced length of the droplet provide an improved means of characterizing droplet on fiber systems. Experiments are reported showing the changes in contact length, droplet height, and inflection angle for poly(dimethyl)siloxane oils on copper cylinders of different diameters. These cylinders are produced from the same initial copper wire by etching in sodium hydroxide to produce controlled diameters ranging from 0.07 to 0.49 mm. As the curvature increases with reducing diameter the influence of gravity diminishes and the shape increasingly conforms to a symmetric barreling droplet type. Furthermore, as the reduced volume of fluid increases the inflection angle increases from 7° to 30° while the contact angle remains at 0°. Consistency between measured values of equilibrium parameters are compared to the theoretical values which we compute numerically and the suggested radius and volume dependence of the inflection angle is confirmed.