An angular extension at gamma-ray energies of 52 arcseconds is detected for the Crab nebula, revealing the emission region of the highest-energy gamma rays; simulations of the electromagnetic emission provide a non-trivial test of our understanding of particle acceleration in the Crab nebula. The Crab nebula is one of the most-studied cosmic particle accelerators, shining brightly across the entire electromagnetic spectrum up to very-high-energy gamma rays(1,2). It is known from observations in the radio to gamma-ray part of the spectrum that the nebula is powered by a pulsar, which converts most of its rotational energy losses into a highly relativistic outflow. This outflow powers a pulsar wind nebula, a region of up to ten light-years across, filled with relativistic electrons and positrons. These particles emit synchrotron photons in the ambient magnetic field and produce very-high-energy gamma rays by Compton up-scattering of ambient low-energy photons. Although the synchrotron morphology of the nebula is well established, it has not been known from which region the very-high-energy gamma rays are emitted(3-8). Here we report that the Crab nebula has an angular extension at gamma-ray energies of 52 arcseconds (assuming a Gaussian source width), much larger than at X-ray energies. This result closes a gap in the multi-wavelength coverage of the nebula, revealing the emission region of the highest-energy gamma rays. These gamma rays enable us to probe a previously inaccessible electron and positron energy range. We find that simulations of the electromagnetic emission reproduce our measurement, providing a non-trivial test of our understanding of particle acceleration in the Crab nebula.