We report a comprehensive ARPES study of NaFeAs, a prototypical parent compound of the Fe-based superconductors. By mechanically detwinning the samples, we show that in the nematic phase (below the structural transition at Ts = 54 K but above the antiferromagnetic transition at TN = 43 K) spectral weight is detected on only the elliptical electron pocket along the longer a orth axis. This dramatic anisotropy is likely to arise as a result of coupling to a fluctuating antiferromagnetic order in the nematic phase. In the long-range ordered antiferromagnetic state below TN , this single electron pocket is backfolded and hybridises with the hole bands, leading to the reconstructed Fermi surface. By careful analysis of the kz variation, we show that the backfolding of spectral weight in the magnetic phase has a wavector of (π,0,π), with the c-axis component being in agreement with the magnetic ordering in NaFeAs observed by neutron scattering. Our results clarify the origin of the tiny Fermi surfaces of NaFeAs at low temperatures and highlight the importance of the three-dimensional aspects of the electronic and magnetic properties of Fe-based superconductors.The dome of unconventional superconductivity in the phase diagrams of Fe-based superconductors usually appears once the stripe antiferromagnetic order found in the parent compounds is suppressed. This proximity has inspired many works which use a spin-fluctuation pairing mechanism to explain the relatively high T c superconductivity in these materials [1,2]. However this picture is challenged by the existence of a distinct "nematic" phase in compounds such as NaFeAs, characterised by a tetragonal-orthorhombic structural transition of the lattice and twofold symmetry of the electronic structure, but without static magnetic order [3,4]. This led to some suggestions that there is a separate symmetry-breaking instability of orbital order [5], which was supported by an apparently large energy scale for d xz −d yz orbital splitting in detwinned ARPES measurements of BaFe 2 As 2 [6] and NaFeAs [7,8]. However the origin of the nematic phase is still not settled, with other groups proposing a magnetically-driven 'spinnematic' state characterised by strong antiferromagnetic fluctuations with a chosen direction but finite magnetic correlation length [9][10][11]. Thus it is important to continue to look for new insights in the parent compounds of the Fe-based superconductors, especially in systems where the nematic and magnetic phases can be distinguished such as NaFeAs.NaFeAs is a prototypical parent compound of the Fe-based superconductors, with a tetragonal-orthorhombic structural transition at T s = 54 K, and a magnetic transition at T N = 43 K. A superconducting dome reaching a maximum T c ≈ 20 K is found upon doping with Co [9, 12], Ni [13] or Rh [14] on the Fe site. While not exhibiting ordering temperatures as high as BaFe 2 As 2 (T s ≈ T N ≈ 134 K [15]) or LaFeAsO (T s ≈ 165 K, T N ≈ 145 K [16]), it offers a convenient test bed for ideas about the nematic phase due ...