1. Given a square pyramid with a base edge length of 2 and a slant height of 3, the radius of its inscribed sphere is $\qquad$

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Answer: $\frac{2 \sqrt{14}-\sqrt{7}}{7}$
Difficulty: Easy
Assessment: Solid Geometry, Inscribed Sphere
Analysis: Consider the volume of the pyramid $V=\frac{1}{3} \times 2 \times 2 \times \sqrt{3^{2}-2}=\frac{4 \sqrt{7}}{3}$
Also, $V=\frac{1}{3} r\left(4 S_{\text {side }}+S_{\text {base }}\right)=r \frac{8 \sqrt{2}+4}{3} \Rightarrow r=\frac{2 \sqrt{14}-\sqrt{7}}{7}$