1. **Problem statement:** We need to find the size of angle $\theta$ in a cuboid where the edges meeting at $\theta$ are 3 mm and 14 mm, and the opposite edge is 8 mm. 2. **Understanding the problem:** The angle $\theta$ is formed between the edges of lengths 3 mm and 14 mm. The 8 mm edge is opposite to this angle, suggesting a right triangle formed by these edges. 3. **Formula used:** To find the angle between two edges in a right triangle, we use the cosine rule: $$\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a$ and $b$ are the sides forming the angle, and $c$ is the opposite side. 4. **Assign values:** Let $a = 3$, $b = 14$, and $c = 8$. 5. **Calculate cosine of $\theta$:** $$\cos(\theta) = \frac{3^2 + 14^2 - 8^2}{2 \times 3 \times 14} = \frac{9 + 196 - 64}{84} = \frac{141}{84} = 1.6786$$ 6. **Check the value:** Since cosine values must be between -1 and 1, $1.6786$ is invalid, indicating the angle is not between edges 3 mm and 14 mm directly. 7. **Reconsider the triangle:** The angle $\theta$ is between the 3 mm and 14 mm edges, and the 8 mm edge is the vertical height. The triangle formed is right-angled with sides 3 mm and 14 mm adjacent to $\theta$, and the hypotenuse is the diagonal on the base. 8. **Calculate the diagonal on the base:** $$d = \sqrt{3^2 + 14^2} = \sqrt{9 + 196} = \sqrt{205} = 14.3178$$ 9. **Calculate $\theta$ using tangent:** Since $\theta$ is between 3 mm and 14 mm edges, we can use: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{14}$$ 10. **Calculate $\theta$:** $$\theta = \tan^{-1}\left(\frac{3}{14}\right) = \tan^{-1}(0.2143) = 12.2^\circ$$ **Final answer:** $$\boxed{\theta = 12.2^\circ}$$