1. **State the problem:** We need to find the area of a regular hexagon given its apothem length $10\sqrt{3}$ cm and the total area $1038$ cm². 2. **Formula for the area of a regular polygon:** $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$$ 3. **Important rules:** - The apothem is the perpendicular distance from the center to a side. - For a regular hexagon, the apothem $a$ relates to the side length $s$ by $a = \frac{\sqrt{3}}{2} s$. 4. **Find the side length $s$ using the apothem:** $$a = 10\sqrt{3} = \frac{\sqrt{3}}{2} s$$ Multiply both sides by 2: $$2 \times 10\sqrt{3} = \cancel{2} \times \frac{\sqrt{3}}{\cancel{2}} s \Rightarrow 20\sqrt{3} = \sqrt{3} s$$ Divide both sides by $\sqrt{3}$: $$\frac{20\sqrt{3}}{\sqrt{3}} = \cancel{\sqrt{3}} s / \cancel{\sqrt{3}} \Rightarrow 20 = s$$ So, the side length $s = 20$ cm. 5. **Calculate the perimeter $P$ of the hexagon:** $$P = 6 \times s = 6 \times 20 = 120 \text{ cm}$$ 6. **Calculate the area using the formula:** $$\text{Area} = \frac{1}{2} \times P \times a = \frac{1}{2} \times 120 \times 10\sqrt{3} = 60 \times 10\sqrt{3} = 600\sqrt{3}$$ 7. **Approximate the area:** $$600\sqrt{3} \approx 600 \times 1.732 = 1039.2 \text{ cm}^2$$ 8. **Compare with given area:** The given area is $1038$ cm², which is very close to our calculated value, confirming the correctness. **Final answer:** The area of the regular hexagon is approximately **1039.2 cm²**.