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Formulae are derived from first principles which predict the wear depth of a ball and a flat surface through time as they slide against each other, in relation to any phenomenological law for wear volume, and taking into account the effect of component geometry. The equations can be fit using experimental wear volume data from ball-on-flat tribometers. The formulae remove previous limiting approximations made in the literature and extend to the prediction of the wear depth of both contacting surfaces. The wear model accords with a previous model that is validated by pin-on-disc testing of a steel/steel contact. The current paper uses the formulae derived to predict the wear depth of a diamond-like carbon (DLC) coating and a steel ball as they slide against each other in deionised water. An Archard equation is used to predict the wear volume of each surface; however, a DLC coating is known to form a transfer layer which reduces the rate of wear, and since this scenario does not obey Archard's law directly, a time-dependent-specific wear rate is used to fit a semi-empirical model to experimental results. The final model predicts the wear depth of the ball and flat accurately. [ABSTRACT FROM AUTHOR]