The article is devoted to the creation of an effective method of mathematical scanning of surfaces, ensuring high accuracy of calculations of their areas. This problem is becoming particularly relevant in the context of the constant growth of requirements for the level of graphic competence of engineers and designers, as well as the active introduction of information technology and automation of project activities. An original method based on dividing a surface into a system of parallel lines and then constructing its sweep on a plane is described. This approach guarantees obtaining a reliable two-dimensional model that adequately reflects the actual configuration of the surface. The procedures for constructing sweeps for both classical deployable surfaces (cone, pyramid) and non-deployable surfaces (sphere, open torus, abstract surface of rotation) are described in detail. The potential application possibilities of the proposed method in industry, architecture and cartography are noted, however, specific practical examples are not given in the article. The results of experimental tests confirming the high accuracy and reliability of the developed methodology are presented. The potential of using this technique in the educational process of technical universities is discussed, where it can significantly enrich the programs of geometric and graphic training of students. The advantages of introducing specialized courses dedicated to new tools of automatic design and spatial modeling are considered. The main conclusions of the article are to identify promising areas for the development of mathematical methods of surface scanning, which will effectively integrate modern information technologies into the processes of professional education and scientific and technological progress. The work is intended for specialists dealing with the theory and practice of design, mathematics and engineering.