In robot-assisted needle-based medical procedures, path planning for a flexible needle is challenging with regard to time consumption and searching robustness for the solution due to the nonholonomic motion of the needle tip and the presence of anatomic obstacles and sensitive organs in the intended needle path. We propose a novel and fast path planning algorithm for a robot-assisted active flexible needle. The algorithm is based on Rapidly-Exploring Random Trees combined with reachability-guided strategy and greedy heuristic strategy. Linear segments are taken into consideration to the paths, and insertion orientations are relaxed by the introduction of the linear segments. The proposed algorithm yields superior results as compared to the commonly used algorithm in terms of computational speed, form of path and robustness of searching ability, which potentially can make it suitable for the real-time intraoperative planning for clinical procedures.