Let P(X,Y,Z) denote a homogeneous polynomial in ℂ[X,Y,Z] such that X_P = {(u : v : w) ∈ ℂℙ² | P(u,v,w) = 0} defines a smooth complex projective curve in ℂℙ². It inherits a Riemannian metric from the Fubini-Study metric on ℂℙ².

Is there an explicit formula or algorithm that gives the area of X_P as a real surface in terms of the polynomial P(X,Y,Z)?

Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\mathbb{C}\mathbb{P}^2$. It inherits a Riemannian metric from the Fubini-Study metric on $\mathbb{C}\mathbb{P}^2$.

Is there an explicit formula or algorithm that gives the area of $X_P$ as a real surface in terms of the polynomial $P(X,Y,Z)$?