In this thesis we constructed the Jordan measure and its properties. At the beginning, in Chapter 1, we have axiomatically constructed a plane and defined the segment, convex set, convex hull and half-plane. We have proved their properties which we will be using in further chapters. In Chapter 2, we have introduced the concept of a triangle as the convex hull of a set which contains three non collinear points, and the concept of a polygon as a union of finite number of triangles. We have defined a border of the triangle, its interior, and observed what is valid for relations between line and triangle, two triangles and their interiors. In Chapter 3, we have developed concepts of complex of segments, segmentation of a set, triangulation of a set and a complex of triangles. Then in Chapter 4, we have introduced functions which calculate area of the triangle and polygon. In the final chapter, Chapter 5, we have defined a number which is called the Jordan measure of a set, considering the polygon area. In this final chapter, we have also studied the properties of the Jordan measure.