The dust density is a central parameter of a dusty plasma. Here, a tomography setup for the determination of the three-dimensionally resolved density distribution of spatially extended dust clouds is presented. The dust clouds consist of micron-sized particles confined in a radio frequency argon plasma, where they fill almost the entire discharge volume. First, a line-of-sight integrated dust density is obtained from extinction measurements, where the incident light from an LED panel is scattered and absorbed by the dust. Performing these extinction measurements from many different angles allows the reconstruction of the 3D dust density distribution, analogous to a computer tomography in medical applications.