9. (21 points) Let the side length of the base of a regular tetrahedron be 1, and the length of the lateral edges be 2. Find its volume and the radius of the inscribed sphere.



保留了源文本的换行和格式。
From the given, the height of the pyramid is
$$
h=\sqrt{2^{2}-\left(\frac{1}{\sqrt{3}}\right)^{2}}=\sqrt{\frac{11}{3}},
$$

its volume $V=\frac{1}{3} \times \frac{\sqrt{3}}{4} h=\frac{\sqrt{11}}{12}$,
its surface area $S=\frac{\sqrt{3}}{4}+\frac{3 \sqrt{15}}{4}$.
Thus, the radius of the inscribed sphere $r=\frac{3 V}{S}=\frac{\sqrt{11}}{\sqrt{3}+3 \sqrt{15}}$.