Let $X$ be a smooth scheme, with a sheaf of graded quasi-coherent algebras $\mathcal{A}^*$, that yields a cone $C$ (in the sense of Fulton's intersection theory). Suppose that $\mathcal{A}^1$ is a rank $r$ locally free sheaf, and denote $\mathcal{E}$ the associated vector bundle.

Is there a way to relate the Segre classes of $C$ and $\mathcal{E}$ in $X$ ?

My motivation is the following: if $\pi : Y\rightarrow X$ is a projective familly, with a relatively ample line bundle $L$ such that $\pi_*L$ is a vector bundle, I would like to compare the Segre classes of $\pi_*L$ with the cone $\bigoplus_n \pi_* L^{\otimes n}$ (adding the relevant hypotheses on $L$).