In this paper, we characterize the aggregate interference power in directional millimeter-wave communications, capturing the effect of beamforming. The analysis considers a general distance-based path loss with Rayleigh and Rician fading channels and a sectored antenna model. Moreover, the nodes are uniformly distributed over a circular or annular area centered at the receiver. The main contribution of the paper is the derivation of the moment generating function (MGF) of the aggregate interference power. The MGF is adopted in a moment-based approximation to a Gamma distribution and used as a model of the aggregate interference power. Several simulations confirm the effectiveness of the proposed approximation for different scenarios, highlighting the effect of directional communications on the aggregate interference power.