The social cost of carbon (SCC), commonly referred to as the carbon price, is the monetized damage from emitting one unit of CO2 to the atmosphere. The SCC is typically obtained from large‐scale computational Integrated Assessment Models (IAMs) that consolidate interdisciplinary climate research inputs to obtain a carbon price estimate relevant for policy‐making. However, the climate economy interactions of IAMs remain inaccessible to scientists in general. Here we develop a simple closed‐form formula that captures the key physical and economic determinants of the SCC in the IAMs. For a mainstream IAM, it explains over 99 percent of the within‐model variation originating from structural uncertainties; in an inter‐model comparison, the structural variation captured by the formula matches closely a SCC distribution of previous SCC estimates. The precise replication of the SCC estimates is strikingly free of details such as those on future policy and technology options, or even carbon concentration levels; the size of the current economy and the emissions‐temperature‐damage response are the dominant SCC determinants in the IAMs. The structural interpretation given allows decision‐makers to disentangle the subjective and structural determinants of the carbon price. Structural uncertainties alone lead to a strongly right‐skewed density with median 15 €/tCO2, mean 31 €/tCO2, and more than 5 percent probability for higher than 100 €/tCO2 for year 2015.