1. **State the problem:** We want to find the total surface area of the region common to the three cylinders given by: $$x^2 + y^2 = 3^2$$ $$x^2 + z^2 = 3^2$$ $$z^2 + y^2 = 3^2$$ 2. **Understand the geometry:** This intersection is known as the Steinmetz solid formed by three mutually perpendicular cylinders of radius 3. 3. **Known result for three cylinders:** The surface area of the Steinmetz solid formed by three cylinders each of radius $r$ is: $$S = 24r^2$$ 4. **Substitute the radius:** Here, $r = 3$, so $$S = 24 imes 3^2 = 24 imes 9 = 216$$ 5. **Conclusion:** The total surface area enclosing the common region is $$\boxed{216}$$ units squared. This formula and result come from the symmetry and integrals of the intersecting cylinders' surfaces.