1. **State the problem:** Find the volume of a cone with radius $r=8$ mm and height $h=10$ mm. 2. **Recall the formula:** The volume $V$ of a cone is given by $$ V = \frac{1}{3} \pi r^2 h $$ 3. **Substitute values:** Using $r=8$ mm and $h=10$ mm, $$ V = \frac{1}{3} \pi \times 8^2 \times 10 $$ 4. **Simplify:** Calculate the square of 8, $$ 8^2 = 64 $$ Then, $$ V = \frac{1}{3} \pi \times 64 \times 10 = \frac{1}{3} \pi \times 640 $$ 5. **Calculate numerically:** $$ V \approx \frac{1}{3} \times 3.1416 \times 640 \approx 670.21 $$ 6. **Round the answer:** The volume rounded to the nearest hundredth is $$ 670.21 \text{ cubic millimeters} $$