Small-diameter shape sensors have a variety of potential applications in medicine such as the positioning of breast tumor chest needles, shape display of intestinal endoscopes, and positioning of cardiac vascular catheters. The shape-reconstruction and end-positioning accuracies of shape sensors have always been major concerns of researchers. To improve the shape-reconstruction accuracy of a fiber Bragg grating (FBG) shape sensor, the strain transfer law of an FBG shape sensor with 90° distribution of fiber around the center of the substrate was studied and verified experimentally. Epoxy resin glues the FBG to the surface of the nickel-titanium alloy wire to form a four-layer structure of the substrate-bonding layer-fiber-bonding layer of the FBG shape sensor. Based on a simplified model of strain transmission mechanics of the FBG sensor, the corresponding average strain transfer rate formula was derived. The effects of the fiber-to-substrate center distance, bond layer length, thickness, and elastic modulus on the average strain transfer rate of FBG sensors were also analyzed and experimentally verified. Experimental results show that the average strain transmission rate of the packaged FBG sensor increases with increasing substrate length in the effective range, whereas an increase in the outer diameter of the bond layer has little effect on the average strain transmission rate, thus verifying the applicability of the theoretical model. The strain transfer rate is introduced into the shape reconstruction, and the shape-reconstruction accuracy of the FBG shape sensor is reduced from 3.5% to 2.7%.