The surface-brightness parameter F is calibrated in terms of the Strömgren intermediate-band colour b−y. The relation F−(b−y)o valid for Cepheids is calibrated using accurate near-infrared radii and distances for selected Cepheids. We have obtained uvby photometry for non-Cepheid giant and supergiant stars with known angular diameters and compared the slope and zero-point of their F−(b−y)o relation with the Cepheid calibration. We found that the two calibrations are significantly different. The theoretical models lie in between the two calibrations. It is remarked that F-colour relations derived from non-Cepheids and involving blue colours (e.g. B−V or b−y) are not applicable to Cepheids, while those involving redder colours (e.g. V−R, V−K or V−J) also produce good radii for Cepheids. Selected Cepheids as calibrators lead to the accurate relation F = 3.898(±0.003)−0.378(±0.006)(b−y)o, which allowed the calculation of radii and distances for a sample of 59 Galactic Cepheids. The uncertainties in the zero-point and slope of the above relation are similar to those obtained from near-infrared colours, and determine the accuracies in radii and distance calculations. While infrared light and colour curves for Cepheids may be superior in precision, the intermediate-band b−y colour allows the recovery of mean radii with an accuracy comparable to those obtained from the infrared solutions. The derived distances are consistent within the uncertainties with those predicted by a widely accepted period-luminosity relationship. Likewise, the resulting period-radius relation from the intermediate-band approach is in better agreement with infrared versions than with optical versions of this law. It is highlighted that the intermediate-band calibration of the surface-brightness method in this work is of comparable accuracy to the near-infrared calibrations. The present results stress the virtues of uvby in determining the physical parameters of supergiant stars of intermediate temperature.