  We prove sharp regularity estimates for solutions of obstacle type problems
driven by a class of degenerate fully nonlinear operators; more specifically,
we consider viscosity solutions of \[ |D u|^\gamma F(x, D^2u) =
f(x)\chi_{\{u>\phi\}} \textrm{ in } B_1 \] with $\gamma>0$, $\phi \in C^{1,
\alpha}(B_1)$ for some $\alpha\in(0,1]$ and $f\in L^\infty(B_1)$ constrained to
satisfy \[ u\geq \phi\textrm{ in } B_1 \] and prove that they are
$C^{1,\beta}(B_{1/2})$ (and in particular along free boundary points) where
$\beta=\min\left\{\alpha, \frac{1}{\gamma+1}\right\}$. Moreover, we achieve
such a feature by using a recently developed geometric approach which is a
novelty for these kind of free boundary problems. Further, under a natural
non-degeneracy assumption on the obstacle, we prove that the free boundary
$\partial\{u>\phi\}$ has zero Lebesgue measure. Our results are new even for
seemingly simple model as follows \[
  |Du|^\gamma \Delta u=\chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. \]
