Abstract: The operating envelope of distribution networks decouples the safe operation of distribution networks from the regulation of distributed energy resources (DER), and maps the safe-operation constraints of distribution networks into DER output constraints. This provides an effective means to solve the problem of different ownership entities and regulation objectives between distribution networks and DER. The key to designing operating envelopes is to calculate the allowable fluctuation range of power at each node. Existing researches mainly focus on the active power operating envelopes while ignoring the reactive power fluctuation range. This paper studies the calculation method for the active-reactive power operating envelope of distribution networks. First, the rotating rectangle model of active-reactive power operating envelopes is constructed, and the active-reactive power region in operating envelopes is enlarged by introducing the rotating parameter. Second, the robust optimization model of operating envelope calculation is established considering voltage and power flow constraints, and its quadratic- constrained programming problem is derived. Then, a fast calculation algorithm of operating envelopes is proposed based on the network topology and parameter features, thus significantly improving the computational efficiency of solving optimization problems. Finally, the effectiveness of the proposed calculation method for distribution network operating envelopes is verified on different distribution networks.