1. **State the problem:** Find the volume of a cylinder with radius $r=5$ mm and height $h=25.5$ mm. 2. **Formula:** The volume $V$ of a cylinder is given by $$V=\pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Substitute values:** $$V=\pi \times 5^2 \times 25.5$$ 4. **Calculate:** $$V=\pi \times 25 \times 25.5$$ 5. **Simplify:** $$V=637.5\pi$$ cubic millimeters (exact volume in terms of $\pi$). 6. **Approximate:** Using $\pi \approx 3.1416$, $$V \approx 637.5 \times 3.1416 = 2001.8$$ cubic millimeters. **Final answer:** The volume of the cylinder is $$637.5\pi$$ cubic millimeters or approximately $$2001.8$$ cubic millimeters.