Three-dimensional electron gas is the key to the excellent performance of polarization-doped field-effect transistors (PolFETs), but no quantitative theoretical work has been conducted on it. In this article, a quantitative theory of the three-dimensional electron gas is presented. Based on a quantitative study on the polarization effect of the AlGaN layer of graded Al components in a PolFET, the volume polarization charge distribution is calculated, and the depletion and neutral approximations of the AlGaN layer are proposed in combination with the space charge analysis. The Fermi-Dirac statistics and energy band-potential relationship are introduced to establish equations for solving the depletion width. The boundary conditions and approximation are defined, and the numerical and analytical solutions of the equation are given (the maximum relative error is less than 3%).