1. **State the problem:** We need to find the length of the diagonal connecting opposite vertices of a cuboid with dimensions 1 cm, 4 cm, and 3 cm. 2. **Formula:** The diagonal $d$ of a cuboid with length $l$, width $w$, and height $h$ is given by the 3D Pythagorean theorem: $$d = \sqrt{l^2 + w^2 + h^2}$$ 3. **Substitute the values:** Here, $l=3$, $w=4$, and $h=1$. $$d = \sqrt{3^2 + 4^2 + 1^2}$$ 4. **Calculate the squares:** $$d = \sqrt{9 + 16 + 1}$$ 5. **Sum inside the square root:** $$d = \sqrt{26}$$ 6. **Evaluate the square root:** $$d \approx 5.099$$ 7. **Round to 3 significant figures:** $$d \approx 5.10$$ **Final answer:** The length of the diagonal is approximately $5.10$ cm.