We present a study of the shape of the Large Magellanic Cloud (LMC) disk. We use the brightnesses of core helium burning red clump stars identified in V-I, I color-magnitude diagrams of 50 randomly selected LMC fields observed with the Cerro Tololo Inter-American Observatory 0.9 m telescope to measure relative distances to the fields. Random photometric errors and errors in the calibration are controlled to <~1%. Following correction for reddening measured through the color of the red clump, we solve for the inclination and position angle of the line of nodes of the tilted plane of the LMC, finding i=35.8d+/-2.4d and θ=145°+/-4°. Our solution requires that we exclude 15 fields in the southwest of the LMC that have red clump magnitudes ~0.1 mag brighter than the fitted plane. On the basis of these fields, we argue that the LMC disk is warped and twisted, containing features that extend up to 2.5 kpc out of the plane. We argue that alternative ways of producing red clump stars brighter than expected, such as variations in age and metallicity of the stars, are unlikely to explain our observations.