Catastrophic collapses of steep volcano flanks threaten many populated regions, and understanding factors that promote collapse could save lives and property. Large collapses of hydrothermally altered parts of Mount Rainier have generated far-traveled debris flows; future flows would threaten densely populated parts of the Puget Sound region. We evaluate edifice collapse hazards at Mount Rainier using a new three-dimensional slope stability method incorporating detailed geologic mapping and subsurface geophysical imaging to determine distributions of strong (fresh) and weak (altered) rock. Quantitative three-dimensional slope stability calculations reveal that sizeable flank collapse (>0.1 km3) is promoted by voluminous, weak, hydrothermally altered rock situated high on steep slopes. These conditions exist only on Mount Rainier's upper west slope, consistent with the Holocene debris-flow history. Widespread alteration on lower flanks or concealed in regions of gentle slope high on the edifice does not greatly facilitate collapse. Our quantitative stability assessment method can also provide useful hazard predictions using reconnaissance geologic information and is a potentially rapid and inexpensive new tool for aiding volcano hazard assessments.