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Fan-beam scan is a configuration widely used in clinical computed tomography (CT) because it is easy to implement and control. Filtered-backprojection (FBP)-based algorithms have been developed previously for exact reconstruction of the entire image or an ROI within the image from the data acquired in fan-beam scans. Recently, we have developed a 3D based algorithm for image reconstruction on PI-line segments in a helical cone-beam scan. In this work, we demonstrated that the 3D FBP-based algorithm indeed provided a rather general FBP-based formulation for image reconstruction. Based upon this formulation we derived new FBP-based algorithms for image reconstruction in the fan-beam scans, which can be interpreted as the special case of the helical scan. Existing algorithms corresponding to the new derived algorithms were identified. We also performed a preliminary numerical study to verify our theoretical results. The results in the work can readily be generalized to other non-circular trajectories.