如图, 已知正三棱雉 $S-A B C$ 的侧面积是底面积的 2 倍, 正三棱雉的高 $S O=3$, 求此正三棱雉的表面积.

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, 设正三棱雉的底面边长为 $a$, 斜高为 $h^{\prime}$. 过点 $O$ 作 $O E \perp A B$, 与 $A B$ 交于点 $E$, 连接 $S E$, 则 $S E \perp A B$,
$S E=h^{\prime}$.

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$\because S_{\text {侧 }}=2 S_{\text {底 }}, \therefore \frac{1}{2} \times 3 a \times h^{\prime}=\frac{\sqrt{3}}{4} a^{2} \times 2, \therefore a=\sqrt{3} h^{\prime}$.

$\because S O \perp O E, \therefore S O^{2}+O E^{2}=S E^{2}$,

$\therefore 3^{2}+\left(\frac{\sqrt{3}}{6} \times \sqrt{3} h^{\prime}\right)^{2}=h^{\prime} \quad 2, \quad \therefore h^{\prime}=2 \sqrt{3}, \quad \therefore a=\sqrt{3} h^{\prime}=6$.

$\therefore S_{\text {底 }}=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 6^{2}=9 \sqrt{3}, S_{\text {侧 }}=2 S_{\text {底 }}=18 \sqrt{3}$,

$\therefore S_{\text {表 }}=S_{\text {侧 }}+S_{\text {底 }}=18 \sqrt{3}+9 \sqrt{3}=27 \sqrt{3}$.