  A general approach to compute the spherical measure of submanifolds in
homogeneous groups is provided. We focus our attention on the homogeneous
tangent space, that is a suitable weighted algebraic expansion of the
submanifold. This space plays a central role for the existence of blow-ups.
Main applications are area-type formulae for new classes of $C^1$ smooth
submanifolds. We also study various classes of distances, showing how their
symmetries lead to simpler area and coarea formulas. Finally, we establish the
equality between spherical measure and Hausdorff measure on all horizontal
submanifolds.
