1. **State the problem:** We need to find the volume of a sphere with radius $r = 4$ mm. 2. **Formula:** The volume $V$ of a sphere is given by $$V = \frac{4}{3} \pi r^3$$ where $\pi$ is approximately 3.14. 3. **Substitute the values:** $$V = \frac{4}{3} \times 3.14 \times 4^3$$ 4. **Calculate the cube of the radius:** $$4^3 = 4 \times 4 \times 4 = 64$$ 5. **Plug in the cube value:** $$V = \frac{4}{3} \times 3.14 \times 64$$ 6. **Multiply $3.14$ and $64$:** $$3.14 \times 64 = 200.96$$ 7. **Calculate the volume:** $$V = \frac{4}{3} \times 200.96$$ 8. **Multiply numerator and denominator:** $$V = \frac{4 \times 200.96}{3} = \frac{803.84}{3}$$ 9. **Divide to simplify:** $$V = \cancel{\frac{803.84}{3}} = 267.95$$ 10. **Round to the nearest hundredth:** $$V \approx 267.95$$ cubic millimeters. **Final answer:** The volume of the sphere is approximately **267.95 cubic millimeters**.