3. If a square pyramid with a base edge length of 2 is inscribed with a sphere of radius $\frac{1}{2}$, then the volume of this square pyramid is .. $\qquad$
3. $\frac{16}{9}$.

Let $\theta$ be the dihedral angle between the base and a side face of a regular quadrilateral pyramid. Then $\tan \frac{\theta}{2}=\frac{1}{2}$.
Thus, $\tan \theta=\frac{2 \tan \frac{\theta}{2}}{1-\tan ^{2} \frac{\theta}{2}}=\frac{4}{3}$. Therefore, the height of the regular quadrilateral pyramid is $1 \times \tan \theta=\frac{4}{3}$.
Hence, the volume of this quadrilateral pyramid is
$$
\frac{1}{3} \times 2^{2} \times \frac{4}{3}=\frac{16}{9} .
$$