1. **State the problem:** Find the area of a regular polygon given the radius (or apothem) of 15 cm. 2. **Formula for the area of a regular polygon:** $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$$ 3. **Important notes:** - The apothem is the perpendicular distance from the center to a side. - The radius is the distance from the center to a vertex. - If only the radius is given, we need the number of sides or the apothem to find the area. 4. **Given:** Radius (or apothem) = 15 cm. 5. **Missing information:** Number of sides or apothem length if 15 cm is radius. Since the problem states "with the given radius or apothem 15 cm" but does not specify the number of sides, we cannot find the exact area without the number of sides. **Assuming the polygon is an equilateral triangle (3 sides) for demonstration:** 6. For an equilateral triangle, the apothem $a$ relates to the radius $R$ by: $$a = R \cos(\pi/3) = 15 \times \cos(60^\circ) = 15 \times 0.5 = 7.5 \text{ cm}$$ 7. The side length $s$ of the triangle is: $$s = 2 R \sin(\pi/3) = 2 \times 15 \times \sin(60^\circ) = 30 \times \frac{\sqrt{3}}{2} = 15\sqrt{3} \text{ cm}$$ 8. The perimeter $P$ is: $$P = 3s = 3 \times 15\sqrt{3} = 45\sqrt{3} \text{ cm}$$ 9. Calculate the area: $$\text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 45\sqrt{3} \times 7.5 = \frac{1}{2} \times 45 \times 7.5 \times \sqrt{3}$$ 10. Simplify: $$= 22.5 \times 7.5 \times \sqrt{3} = 168.75 \sqrt{3} \approx 168.75 \times 1.732 = 292.4 \text{ cm}^2$$ **Final answer:** The area of the regular triangle with radius 15 cm is approximately **292.4 cm\textsuperscript{2}**. If the polygon has a different number of sides, please provide that for an exact answer.