Accurate calculation of beamwidth is essential for optimizing the performance of antenna arrays. However, traditional methods for calculating beamwidth require long computation time or result in low accuracy. To address this issue, we propose the Second-order Taylor Expansion-Approximation (STE-A) method for fast and accurate calculation of beamwidth for non-uniform linear arrays. We evaluate the effectiveness of our proposed method through simulations using MATLAB, and the results demonstrate its efficiency and accuracy. Compared with Fourth-order Taylor Expansion (FTE) method, the relative error of the beamwidth of the proposed method was reduced from 4% to 0.2%, which is comparable in accuracy to the Search method. Additionally, compared to the Search method, the proposed method saves 99% of the calculation time.