I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a point $p\in \mathbb{CP}^n$ then then mean curvature of the distance sphere centered at $p$ with radius $r$ is $$ H(r)=2(n-1)\cot(r)+2\cot(2r) $$ If we use $A(t, \theta)$ to denoted the area element of the distance sphere. Then $$ A^{-1}\frac{dA}{dt}=H(t)$$ Hence we get $$A(t)=2\sin^{2n-1}(t)\cos(t)$$ Therefore we can calculate the volume of $B(p, r)$ as: $$ Vol(B(p, r))=\int_0^r(\int_{\theta\in S^{2n-1}}A(t,\theta)dS_{2n-1})dt $$ $$ =\omega_{2n-1}\int_0^r2\sin^{2n-1}(t)\cos(t)dt $$ where $\omega_{2n-1}$ is the volume of the round sphere $S^{2n-1}$. To get the whole volume we let $r=\pi/2$. whence $$ V(\mathbb{CP}^n, g_{FS})=\omega_{2n-1}\frac{1}{n}=\frac{2\pi^n}{(n-1)!}\frac{1}{n} $$ $$=2\frac{\pi^n}{n!}$$
If we take $n=1$, then clearly $\mathbb{CP}^1$ is the round sphere with radius $1/2$ hence must have volume $\pi$, which is $1/4$ of the volume of unit 2-sphere. However the above formula give $\omega_{2-1}=\omega_{1}=2\pi$.
So does anybody know what's wrong with my calculations? Thanks in advance.