1. **State the problem:** Find the area of a regular pentagon with a radius (distance from center to vertex) of 10 mm. 2. **Formula:** The area $A$ of a regular polygon with $n$ sides and radius $R$ (distance from center to vertex) is given by: $$A = \frac{1}{2} n R^2 \sin\left(\frac{2\pi}{n}\right)$$ 3. **Identify values:** For a pentagon, $n=5$ and $R=10$ mm. 4. **Calculate the central angle:** $$\frac{2\pi}{5} = \frac{2 \times 3.1416}{5} = 1.2566 \text{ radians}$$ 5. **Calculate sine:** $$\sin(1.2566) \approx 0.9511$$ 6. **Plug values into formula:** $$A = \frac{1}{2} \times 5 \times 10^2 \times 0.9511 = \frac{1}{2} \times 5 \times 100 \times 0.9511$$ 7. **Simplify:** $$A = \frac{1}{2} \times 500 \times 0.9511 = 250 \times 0.9511 = 237.775$$ 8. **Round to nearest tenth:** $$A \approx 237.8 \text{ mm}^2$$ **Final answer:** The area of the regular pentagon is approximately **237.8 mm²**.