  We combine state-of-the-art large-scale first principles calculations with a
low-energy continuum model to describe the nearly flat bands of twisted bilayer
graphene at the first magic angle $\theta =1.08^\circ$. We show that the energy
width of the flat band manifold, as well as the energy gap separating it from
the valence and conduction bands, can be obtained only if the out-of-plane
relaxations are fully taken into account. The results agree both qualitatively
and quantitatively with recent experimental outcomes.
