We study nonlinear approximation in Lp(R2),0<p<∞, from n-term rational functions. Our main result relates n-term rational approximation in Lp to nonlinear approximation from a broad class of piecewise polynomials over multilevel triangulations allowing a lot of flexibility and, in particular, arbitrarily sharp angles. This relationship and the existing estimates for spline approximation give a Jackson estimate for n-term rational approximation in terms of a minimal smoothness norm over a large collection of anisotropic Besov-type spaces (B-spaces).