1. **State the problem:** Find the area of a regular hexagon with a given apothem (distance from center to a side) or radius (distance from center to a vertex) of $12\sqrt{3}$ cm. 2. **Identify the formula:** The area $A$ of a regular polygon with $n$ sides, side length $s$, and apothem $a$ is given by: $$A = \frac{1}{2} \times n \times s \times a$$ For a regular hexagon, $n=6$. 3. **Important rules:** - The apothem $a$ is the perpendicular distance from the center to a side. - The radius $R$ is the distance from the center to a vertex. - For a regular hexagon, the radius $R$ equals the side length $s$. - The apothem $a$ relates to the radius $R$ by $a = R \cos(\pi/6) = R \times \frac{\sqrt{3}}{2}$. 4. **Given:** The dashed segment from the center to a vertex is $12\sqrt{3}$ cm, so $R = 12\sqrt{3}$ cm. 5. **Calculate the apothem $a$:** $$a = R \times \frac{\sqrt{3}}{2} = 12\sqrt{3} \times \frac{\sqrt{3}}{2} = 12 \times \frac{3}{2} = 18$$ 6. **Side length $s$:** For a regular hexagon, $s = R = 12\sqrt{3}$ cm. 7. **Calculate the area:** $$A = \frac{1}{2} \times 6 \times s \times a = 3 \times 12\sqrt{3} \times 18$$ 8. **Simplify:** $$3 \times 12 \times 18 \times \sqrt{3} = 3 \times 216 \times \sqrt{3} = 648 \sqrt{3}$$ 9. **Rounded to the nearest tenth:** $$648 \sqrt{3} \approx 648 \times 1.732 = 1122.6$$ **Final answer:** $$\boxed{648 \sqrt{3} \text{ cm}^2 \approx 1122.6 \text{ cm}^2}$$