Given $d\in \mathbb{Z}_{\geq 2}$, for every $\kappa=(k_1,\dots,k_n) \in \mathbb{Z}^{n}$ such that $k_i\geq 1-d$ and $k_1+\dots+k_n=-2d$, denote by $\Omega^d\mathcal{M}_{0,n}(\kappa)$ and $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$ the corresponding stratum of $d$-differentials in genus $0$ and its projectivization respectively. We show that the incidence variety compactification of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$ is isomorphic to the blow-up of $\overline{\mathcal{M}}_{0,n}$ along a specific sheaf of ideals. Along the way we obtain an explicit divisor representing the tautological line bundle on the incidence variety. In the case where none of the $k_i$ is divisible by $d$, the self-intersection number of this divisor computes the volume of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$. We prove a recursive formula which allows one to compute the volume of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$ from the volumes of other strata of lower dimensions. As an application of this formula, we give a new proof of the Kontsevich's formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya-Eskin-Zorich. In another application, we show that the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles at the singularities is given by a piecewise polynomial continuous function of the angles, provided none of the angles is an integral multiple of $2\pi$. We also show that the polynomial expressions of this function are always equal to $\pi^{n-2}$ times a polynomial with rational coefficients and degree at most $n-3$, where $n$ is the number of singularities. This generalizes the results of McMullen and Koziarz-Nguyen on the volumes of the moduli spaces of flat surfaces in genus 0 with convex conical singuliarities.< Réduire