In 1908, the mathematician Felix Klein published a book Elementary Mathematics from an Advanced Standpoint: Geometry. This title aptly characterizes the focus of this thesis. This thesis introduces the axioms for Euclidean and projective plane geometry. Afterwards an arithmetic of lengths, based solely on these axioms, is constructed. By establishing a connection between the geometric axioms and the field axioms, it is demonstrated that these lengths form a field. It is shown that the original geometric plane is isomorphic to the Cartesian plane of lengths. Additionally, the thesis highlights the direct relation between two geometric propositions, Pappos’ theorem and Desargues’ theorem, and commutativity and associativity of the induced field of lengths.