The radiative forcing caused by a volcanic eruption is dependent on several eruption source parameters such as the mass of sulfur dioxide (SO2) emitted, the eruption column height, and the eruption latitude. General circulation models with prognostic aerosol and chemistry schemes can be used to investigate how each parameter influences the volcanic forcing. However, the range of multidimensional parameter space that can be explored is restricted because such simulations are computationally expensive. Here we use statistical emulation to explore the radiative impact of eruptions over a wide covarying range of SO2 emission magnitudes, injection heights, and eruption latitudes based on only 30 simulations. We use the emulators to build response surfaces to visualize and predict the sulfate aerosol e‐folding decay time, the stratospheric aerosol optical depth, and net radiative forcing of thousands of different eruptions. We find that the volcanic stratospheric aerosol optical depth and net radiative forcing are primarily determined by the mass of SO2 emitted, but eruption latitude is the most important parameter in determining the sulfate aerosol e‐folding decay time. The response surfaces reveal joint effects of the eruption source parameters in influencing the net radiative forcing, such as a stronger influence of injection height for tropical eruptions than high‐latitude eruptions. We also demonstrate how the emulated response surfaces can be used to find all combinations of eruption source parameters that produce a particular volcanic response, often revealing multiple solutions. Key Points We demonstrate the feasibility and value of using statistical emulation to quantify the radiative impact of volcanic eruptions Emulated response surfaces illustrate the dependencies of model output such as net radiative forcing on eruption source parameters Emulated response surfaces can also be used to constrain the eruption source parameters for a particular volcanic response