Extreme storms generate ocean waves, with significant heights that may exceed 20 m. Possible climate trends for these events are unknown as they only cover small ocean areas, often missed by nadir-looking satellite altimeters. Because storm waves radiate as swell across ocean basins, they are resolved by the Surface Water and Ocean Topography (SWOT) satellite mission. We find a very sharp decay of swell heights as a function of distance from the storm, which we link to the low frequency shape of the wave spectrum in the storm, quantified by a storm peak period. This period is a useful metric for sizing storms and their associated swells. With this data, SWOT illuminates the investigation of air-sea interactions, coastal impacts and the interpretation of seismic data.