In Euclidean relational particle mechanics, only relative times, relative angles and relative separations are meaningful. Barbour–Bertotti (1982 Proc. R. Soc. Lond. A 382 295) theory is of this form and can be viewed as a recovery of (a portion of) Newtonian mechanics from relational premises. This is of interest in the absolute versus relative motion debate and also shares a number of features with the geometrodynamical formulation of general relativity, making it suitable for some modelling of the problem of time in quantum gravity. I also study similarity relational particle mechanics (‘dynamics of pure shape’), in which only relative times, relative angles and ratios of relative separations are meaningful. This I consider first as it is simpler, particularly in 1 and 2D, for which the configuration space geometry turns out to be well known, e.g. for the ‘triangleland’ (3-particle) case that I consider in detail. Second, the similarity model occurs as a sub-model within the Euclidean model: that admits a shape-scale split. For harmonic oscillator like potentials, similarity triangleland model turns out to have the same mathematics as a family of rigid rotor problems, while the Euclidean case turns out to have parallels with the Kepler–Coulomb problem in spherical and parabolic coordinates. Previous work on relational mechanics covered cases where the constituent subsystems do not exchange relative angular momentum, which is a simplifying (but in some ways undesirable) feature paralleling centrality in ordinary mechanics. In this paper I lift this restriction. In each case I reduce the relational problem to a standard one, thus obtain various exact, asymptotic and numerical solutions, and then recast these into the original mechanical variables for physical interpretation.