1. **Problem:** Find the area of a regular decagon with an apothem of 15. Round your answer to the nearest tenth. 2. **Formula:** The area $A$ of a regular polygon is given by $$A = \frac{1}{2} \times P \times a$$ where $P$ is the perimeter and $a$ is the apothem. 3. **Important rules:** - A regular decagon has 10 equal sides. - The apothem is the perpendicular distance from the center to a side. - To find the perimeter, we need the side length. 4. **Find the side length:** - The central angle of each triangle formed by the apothem is $\frac{360^\circ}{10} = 36^\circ$. - Half of this angle is $18^\circ$. - Using the right triangle formed by the apothem, half side length, and radius: $$\tan(18^\circ) = \frac{\text{half side}}{\text{apothem}} = \frac{s/2}{15}$$ - Solve for $s$: $$s = 2 \times 15 \times \tan(18^\circ)$$ 5. **Calculate side length:** $$s = 30 \times \tan(18^\circ) \approx 30 \times 0.3249 = 9.747$$ 6. **Calculate perimeter:** $$P = 10 \times s = 10 \times 9.747 = 97.47$$ 7. **Calculate area:** $$A = \frac{1}{2} \times 97.47 \times 15 = \frac{1}{2} \times 1462.05 = 731.025$$ 8. **Round to nearest tenth:** $$\boxed{731.0}$$ Final answer: The area of the regular decagon is approximately 731.0 square units.