A measurement of the production of three isolated photons in proton-proton collisions at a centre-of-mass energy root s = 8 TeV is reported. The results are based on an integrated luminosity of 20.2 fb(-1) collected with the ATLAS detector at the LHC. The differential cross sections are measured as functions of the transverse energy of each photon, the difference in azimuthal angle and in pseudorapidity between pairs of photons, the invariant mass of pairs of photons, and the invariant mass of the triphoton system. A measurement of the inclusive fiducial cross section is also reported. Next-to-leading-order perturbative QCD predictions are compared to the cross-section measurements. The predictions underestimate the measurement of the inclusive fiducial cross section and the differential measurements at low photon transverse energies and invariant masses. They provide adequate descriptions of the measurements at high values of the photon transverse energies, invariant mass of pairs of photons, and invariant mass of the triphoton system.