The total harmonic distortion (THD) of a waveform is a standard way to quantify its deviation from a sinusoid. In three-phase systems, we are interested in minimizing 3THD: the component of THD produced by odd nontriplen harmonics. However, its definition involves an infinite sum, making it difficult to evaluate and analyze. This paper solves the problem of finding equivalent closed-form expressions for the 3THD of a staircase waveform. In particular, two expressions are rigorously derived, which reveal 3THD to be a piecewise differentiable function. One expression is shorter but describes the pieces implicitly. The other is longer but describes the pieces explicitly. We minimize 3THD using the closed-form expression and provide a comparison with previous techniques. Finally, we provide experimental results that show close agreement with the theoretical results.