1. **State the problem:** We need to find the area of a regular hendecagon (11-sided polygon) inscribed in a circle, where each side length $s$ is approximately 7.46 millimeters. 2. **Formula for the area of a regular polygon:** The area $A$ of a regular polygon with $n$ sides each of length $s$ is given by $$A = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)}$$ where $n=11$ for a hendecagon. 3. **Substitute the values:** $$A = \frac{11 \times (7.46)^2}{4 \tan\left(\frac{\pi}{11}\right)}$$ 4. **Calculate the numerator:** $$11 \times (7.46)^2 = 11 \times 55.6516 = 611.1676$$ 5. **Calculate the denominator:** $$4 \times \tan\left(\frac{\pi}{11}\right)$$ First, calculate $\frac{\pi}{11} \approx 0.2856$ radians. Then, $$\tan(0.2856) \approx 0.2937$$ So, $$4 \times 0.2937 = 1.1748$$ 6. **Calculate the area:** $$A = \frac{611.1676}{1.1748}$$ 7. **Simplify the fraction:** $$A = \frac{\cancel{611.1676}}{\cancel{1.1748}} = 520.22$$ (rounded to two decimal places) **Final answer:** The area of the regular hendecagon is approximately **520.22 square millimeters**.