# Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $$\mathbb{P}^n$$ with the standard Fubini Study metric.

I searched all over the internet but I can't find an example of a calculation of the volume for a projective subvariety $$X \subset \mathbb{P}^n$$ (we can assume it to be smooth of course).

For example given the projective plane $$\mathbb{P}^2$$ with coordinates $$[x,y,z]$$ and the smooth conic $$C:(xy-z^2=0) \subset \mathbb{P}^2$$ does somebody know how to compute the Volume $$\textit{Vol}(C)$$? Or maybe a reference where a similar computation is carried out? For simplicity we can assume that the ground field is $$\mathbb{C}$$.