SPACE-BASED observations have established that the Sun’s irradiance varies with solar magnetic activity(1-4). A fraction of this variability arises from the increased area of cool gas associated with sunspots, which decreases irradiance; but on average the blocking effect of sunspots is more than offset by the increased emission from photospheric faculae and other effects of magnetic activity(5) (although the precise contributions of these opposing effects remain poorly constrained(6,7)). Here we show that the apparent radius of the Sun, when viewed in the spectral line of neutral iron at 525 nm, also varies in phase with solar magnetic activity. This variation probably results from changes in the temperature profile of the Sun’s atmosphere with the solar cycle. If similar behaviour is found for other spectral lines, changes in the apparent radius of the Sun could account for a significant fraction (similar to 20%) of the total irradiance variations.