Wall roughness induces extra drag in wall-bounded turbulent flows. Mapping any given roughness geometry to its fluid dynamic behaviour has been hampered by the lack of accurate and direct measurements of skin-friction drag. Here, the Taylor–Couette (TC) system provides an opportunity as it is a closed system and allows direct and reliable measurement of the skin-friction. However, the wall curvature potentially complicates the connection between the wall friction and the wall roughness. Here, we investigate a highly turbulent TC flow, with a hydrodynamically fully rough, rotating inner cylinder, while the outer cylinder is kept smooth and stationary. We carry out particle image velocimetry (PIV) measurements in the Twente Turbulent Taylor–Couette (T3C) facility with Reynolds numbers in the range of $4.6\times 10^5 < Re_i < 1.77\times 10^6$. From these we find, while taking into account the influence of the curved walls on the turbulence, that the observed effects of a hydrodynamically fully rough surface are similar for TC turbulence and flat-plate turbulent boundary layer flows (BL). Hence, the equivalent sand grain height ks, that characterizes the drag properties of a rough surface, is similar for both flow geometries. Next, we obtain the dependence of the torque (skin-friction drag) on the Reynolds number for a given wall roughness, characterized by ks, and find agreement with the same results derived from PIV measurements within $5\,\%$. Our findings demonstrate that global torque measurements in the TC facility could be well suited to reliably deduce wall-drag properties for any rough surface.