  We find closed-form solution of the Euler equation for a chiral magnet in
terms of a skyrmion or a meron depending on the relative strengths of magnetic
anisotropy and magnetic field. We show that the relevant length scales for
these solutions primarily depend on the strengths of Dzyaloshinskii-Moriya
interaction through its ratios, respectively, with magnetic field and magnetic
anisotropy. We thus unambiguously determine the parameter dependencies on the
radius of the topological structures particularly of the skyrmions, showing an
excellent agreement with experiments and first-principle studies. An
anisotropic Dzyaloshinskii-Moriya interaction suitable for thin films made with
$C_{nv}$ symmetric materials is found to stabilize anti-skyrmion and
anti-meron, which are prototypical for $D_{2d}$ symmetric systems, depending on
the degree of anisotropy. Based on these solutions, we obtain phase diagram by
comparing the energies of various collinear and non-collinear competing phases.
