Aims. We aim to reproduce the structure of the corona above a solar active region as seen in the extreme ultraviolet (EUV) using a three-dimensional magnetohydrodynamic (3D MHD) model. Methods. The 3D MHD data-driven model solves the induction equation and the mass, momentum, and energy balance. To drive the system, we feed the observed evolution of the magnetic field in the photosphere of the active region AR 12139 into the bottom boundary. This creates a hot corona above the cool photosphere in a self-consistent way. We synthesize the coronal EUV emission from the densities and temperatures in the model and compare this to the actual coronal observations. Results. We are able to reproduce the overall appearance and key features of the corona in this active region on a qualitative level. The model shows long loops, fan loops, compact loops, and diffuse emission forming at the same locations and at similar times as in the observation. Furthermore, the low-intensity contrast of the model loops in EUV matches the observations. Conclusions. In our model the energy input into the corona is similar as in the scenarios of fieldline-braiding or flux-tube tectonics, that is, energy is transported to the corona through the driving of the vertical magnetic field by horizontal photospheric motions. The success of our model shows the central role that this process plays for the structure, dynamics, and heating of the corona.