This work presents a new method to determine a rigid body orientation, based on indirect measurements. This method is driven by the need to minimize elements attached to the observed body and the final cost of the system while preserving the fine accuracy of the determination. The original application of this method is finding in real time a CubeSat orientation during its examination on a testbed. The angular orientation is determined by means of several linear measurements; the number of required measurements, as well as the method of measuring, being different for two variations of the method presented in the paper. One approach is based on measuring the relative positions between the elements of a rotating rigid body and a body with a known orientation. It requires at least 4 sensors. A second approach employs a simple and light target attached to the rotating body and only 3 sensors. Efficiency and limitations of both approaches is estimated by numerical simulations.