Improvements in the approximate formulae for the period of the simple pendulum

This paper is concerned with improvements in some exact formulae for the period of the simple pendulum problem. Two recently presented formulae are re-examined and refined rationally, yielding more accurate approximate periods. Based on the improved expressions here, a particular new formula is proposed for the period. It is shown that the derived formula has considerably high accuracy for all amplitudes of the pendulum, resulting in a maximum percentage relative error of only 0.035.