The critical oxygen partial pressure (Pcrit), typically defined as the PO2 below which an animal's metabolic rate (MR) is unsustainable, is widely interpreted as a measure of hypoxia tolerance. Here, Pcrit is defined as the PO2 at which physiological oxygen supply (α0) reaches its maximum capacity (α; µmol O2 g-1 h-1 kPa-1). α is a species- and temperature-specific constant describing the oxygen dependency of the maximum metabolic rate (MMR=PO2×α) or, equivalently, the MR dependence of Pcrit (Pcrit=MR/α). We describe the α-method, in which the MR is monitored as oxygen declines and, for each measurement period, is divided by the corresponding PO2 to provide the concurrent oxygen supply (α0=MR/PO2). The highest α0 value (or, more conservatively, the mean of the three highest values) is designated as α. The same value of α is reached at Pcrit for any MR regardless of previous or subsequent metabolic activity. The MR need not be constant (regulated), standardized or exhibit a clear breakpoint at Pcrit for accurate determination of α. The α-method has several advantages over Pcrit determination and non-linear analyses, including: (1) less ambiguity and greater accuracy, (2) fewer constraints in respirometry methodology and analysis, and (3) greater predictive power and ecological and physiological insight. Across the species evaluated here, α values are correlated with MR, but not Pcrit. Rather than an index of hypoxia tolerance, Pcrit is a reflection of α, which evolves to support maximum energy demands and aerobic scope at the prevailing temperature and oxygen level., Competing Interests: Competing interests The authors declare no competing or financial interests., (© 2021. Published by The Company of Biologists Ltd.)