Type Ia supernovae, the thermonuclear explosions of white dwarf stars composed of carbon and oxygen, were instrumental as distance indicators in establishing the acceleration of the universe's expansion. However, the physics of the explosion are debated. Here we report a systematic spectral analysis of a large sample of well-observed type Ia supernovae. Mapping the velocity distribution of the main products of nuclear burning, we constrain theoretical scenarios. We find that all supernovae have low-velocity cores of stable iron-group elements. Outside this core, nickel-56 dominates the supernova ejecta. The outer extent of the iron-group material depends on the amount of nickel-56 and coincides with the inner extent of silicon, the principal product of incomplete burning. The outer extent of the bulk of silicon is similar in all supernovae, having an expansion velocity of ∼11,000 kilometers per second and corresponding to a mass of slightly over one solar mass. This indicates that all the supernovae considered here burned similar masses and suggests that their progenitors had the same mass. Synthetic light-curve parameters and three-dimensional explosion simulations support this interpretation. A single explosion scenario, possibly a delayed detonation, may thus explain most type Ia supernovae.