Review of approximate equations for the pendulum period

Precision measurements of the pendulum period as a function of the amplitude can now be made with a variety of instruments including MEMs gyro/accelerometers, and thus theoretical expressions are required for comparison. Unfortunately exact solution of the pendulum equation involves elliptic integrals, which cannot be expressed in terms of elementary functions, and therefore a wide variety of approximations have been published. These range from simple single-term formulae to more sophisticated equations, which apply to a wider range of amplitudes, to an iterative procedure for calculating the precise period. The published approximations are compared as Taylor series expansions, and graphically to indicate their accuracy and their regions of applicability.