We extend classical solutions in contact mechanics to examine the collapse of channels in a soft elastomer. These channels have triangular cross-section and collapse when pressure is applied to the surrounding elastomer. Treating the walls of the channel as indenters that penetrate the channel base, we derive an algebraic mapping between pressure and cross-sectional area. These theoretical predictions are in strong agreement with results that we obtain through finite element analysis and experimental measurements. This is accomplished without data fitting and suggests that the theoretical approach may be generalized to a broad range of cross-sectional geometries in soft microfluidics.