The Arctic sea ice (ASI) area reaches its yearly minimum in early autumn. Satellite measurements show that September ASI area has decreased approximately linearly when considered as a function of global mean temperature (GMT). However, there is no a priori reason to expect the sensitivity to remain constant under further warming. Model projections of September ASI area remain highly uncertain and therefore need to be complemented by observation-based estimates. Following a new approach, we show that the September ASI area is closely proportional to the area outlined by latitudes with zonally averaged January-September temperatures below a critical threshold Tc. We discover a purely geometric effect that leads to accelerating ASI loss with rising temperatures as the September ASI area crosses the outlines of the area defined by Tc. Consistent with Arctic amplification, the zonally averaged temperatures have followed a positive linear trend relative to the global mean, which is projected to continue throughout the century. Linearly extrapolating these zonally averaged temperatures, we infer that the sensitivity of the ASI area to GMT will increase, leading to strongly accelerating melt rates and a complete loss of September ASI for global warming between 1.5 and 2.2 °C above pre-industrial GMT levels, compared to the range of 1.3 to 3.1 °C given by state-of-the-art (CMIP6) models. For the GMT at which ASI is lost, our observation-based projections hence give estimates that are considerably lower, and have a much smaller uncertainty range, compared to model predictions.