High-energy nuclear collisions create a quark-gluon plasma, whose initial condition and subsequent expansion vary from event to event, impacting the distribution of the eventwise average transverse momentum [(Formula presented)]. Disentangling the contributions from fluctuations in the nuclear overlap size (geometrical component) and other sources at a fixed size (intrinsic component) remains a challenge. This problem is addressed by measuring the mean, variance, and skewness of (Formula presented) in (Formula presented) and (Formula presented) collisions at (Formula presented) and 5.44 TeV, respectively, using the ATLAS detector at the LHC. All observables show distinct features in ultracentral collisions, which are explained by a suppression of the geometrical component as the overlap area reaches its maximum. These results demonstrate a new technique to separate geometrical and intrinsic fluctuations, providing constraints on initial conditions and properties of the quark-gluon plasma, such as the speed of sound.