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A clearer and more correct way to do the second calculation is to look at the volume of the preimage of $S_r^{2n+1}$ in standard $S^{2n+3}$. The preimage is a certain torus $T_r$, and one has $$\mathop{Vol} T_r = 2\pi\mathop{Vol} S_r^{2n+1}.$$ At the same time, $$\mathop{Vol} T_r \propto (\cos r)(\sin r)^{2n+1},$$ because $T_r$ is the Riemannian Cartesian product of a circle of radius $\cos r$ and an $S^{2n+1}$ of radius $\sin r$. If you calculate when the derivative of this expression vanishes, you get agreement with your first calculation.

Addendum: You can also directly check that the scale of the $\mathbb{C}P^n$ quotient of $S^{2n+1}_r$, if you want to do the calculation that way, is $\sin r$, which yields a factor of $(\sin r)^{2n}$, in addition to the factor of $\sin 2r \propto (\sin r)(\cos r)$ from the Hopf fiber. So it all fits together.

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A clearer and more correct way to do the second calculation is to look at the volume of the preimage of $S_r^{2n+1}$ in standard $S^{2n+3}$. The preimage is a certain torus $T_r$, and one has $$\mathop{Vol} T_r = 2\pi\mathop{Vol} S_r^{2n+1}.$$ At the same time, $$\mathop{Vol} T_r \propto (\cos r)(\sin r)^{2n+1},$$ because $T_r$ is the Riemannian Cartesian product of a circle of radius $\cos r$ and an $S^{2n+1}$ of radius $\sin r$. If you calculate when the derivative of this expression vanishes, you get agreement with your first calculation.