This work introduces an application of differential geometry to cartography. The mathematical aspects of some geographical projections of Earth surface are revealed together with some of its more important properties. An important problem since the discovery of the 'spherical' form of the Earth is how to compose a reliable map of the surface of the Earth that could prove useful for navigation. In many cartography texts scarce mention is made of the mathematics inherent to mathematical development. At the same time, the few texts on differential geometry (or vector calculus) that consider this problem have some pedagogical deficiencies: (a) the historical origin of the problem is not generally mentioned; (b) concepts are not always defined with clear explanation of their origin and application; (c) formulas are used without much explanation of their derivation. This work attempts to overcome this. The content is easily understandable for a first or second year university student (with a certain degree of knowledge of the calculus of several variables) and it can be used as support in the study of Euclidean space surfaces. The aim is not a thorough study of terrestrial projections, but an overall view of the difficulties that arise by means of the study of some important projections. The name projection derives from the surface of the Earth being projected in different manners, in a plane, a cylinder or a cone (all surfaces with null Gaussian curvature). The relevance of this work lies in a new approach that is used to present some classical projections. This work aims to show how methods from calculus can also be a valid alternative. The intuitive ideas that a student needs for understanding calculus of several variables are reinforced.