10. For a cylinder inscribed in a sphere of radius $R$, the maximum total surface area of the cylinder is

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10. For a cylinder inscribed in a sphere of radius $R$, the maximum total surface area of the cylinder is

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Rongdian $\pi R^{2}(1+\sqrt{5})$.
Analysis Let the radius of the base of the inscribed cylinder be $R \sin \alpha$, then the height is $2 R \cos \alpha$, so the total surface area is
$$
\begin{aligned}
2 \pi(R \sin \alpha)^{2}+2 \pi R \sin \alpha \times 2 R \cos \alpha & =2 \pi R^{2}\left(\sin ^{2} \alpha+\sin 2 \alpha\right)=\pi R^{2}(1-\cos 2 \alpha+2 \sin 2 \alpha) \\
& =\pi R^{2}(1+\sqrt{5} \sin (2 \alpha-\varphi)) \leq \pi R^{2}(1+\sqrt{5})
\end{aligned}
$$

where $\tan \varphi=\frac{1}{2}$, the equality holds when $2 \alpha=\varphi+\frac{\pi}{2}$, hence the maximum value is $\pi R^{2}(1+\sqrt{5})$.