One of the main objectives of topographic surveys has allowed to draw maps or plans of an area of a limited region or terrain, showing physical characteristics of the terrain, such as rivers, lakes, reservoirs, roads, forests, rock formations, ponds, dams, dikes, drainage pits or water supply channels. The accuracy of the measurement will depend on the scale of the map, the method and the instruments utilized. This document provides the mathematical fundamentals of the planimeter, that allows to measure the area of uneven or spherical flat surfaces; this instrument is important in topographic engineering. The knowledge, and the analityc foundation, of this instrument, makes the article not only of a pedagogical nature, but also it provides a historical development depicting its evolution and leading to its digital current version.