We study flat surfaces with conical singularities, their geometry, their deformation spaces and their parameterization. The base surface is a sphere with three holes (pair of pants). We find three sets of parameters of the flat pair of pants with one conical singular point and we describe its deformation space. We introduce a flow which we call Fenchel-Nielsen flow on a deformation space. We study the injectivity of this flow by examining the variation of length functions of geodesic segments or simple closed geodesics along this flow. We study also the parameterization of a flat surface with conical singularities using the lengths of geodesic segments joining singular points or a singular point to a boundary component. At the end of the text, three appendices provide further discussions.
