The paper is devoted to a bolted flanged joint with a flat ring gasket. Simple mathematical models of the flat ring gasket and the flange are formulated. Solutions to the models allowed determining numerically effective shapes of the flat ring and the flange. In the case of the gasket a minimal tension of the bolts was assumed as a criterion, while in the case of the flange the criterion of minimal angle of the flange rotation was applied. Results of the study, shown in the Figures, may serve for practical purposes in designing of pressure vessels and piping.