The large-scale deployment of electric vehicles (EVs) poses critical challenges to the secure and economic operation of power distribution networks (PDNs). Therefore, a method for evaluating the hosting capacity that enables a PDN to determine the EV chargeable area (EVCA) to satisfy the charging demand and ensure the secure operation is proposed in this paper. Specifically, the distribution system operator (DSO) serves as a public entity to manage the integration of EVs by determining the presence of the charging load in the EVCA. Hence, an EVCA optimization model is formulated on the basis of the coupling effect of the charging nodes to determine the range of the available charging power. In this model, nonlinear power flow equations and operational constraints are considered to maintain the solvability of the power flow of the PDN. Subsequently, a novel multipoint approximation technique is proposed to quickly search for the boundary points of the EVCA. In addition, the impact of the demand response (DR) mechanism on the hosting capacity is explored. The results show that the presence of the DR significantly enlarged the EVCA during peak hours, thus revealing the suitability of the DR mechanism as an important supplement to accommodate the EV charging load. The examined case studies demonstrate the effectiveness of the proposed model and show that the unmanaged allocation of the charging load impedes secure operation. Finally, the proposed method provides a reference for the allocation of the EV charging load and a reduction in the risk of line overloading.