Hexagon Garden Area Calculation

The problem asks for the remaining area of a regular hexagon garden after accounting for a circular fountain.

Garden Area Calculation

First, calculate the total area of the regular hexagon garden.

• The formula for the area of a regular hexagon with side length '$s$' is: $A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2$
• Given the side length '$s = 20$' m.
• Substitute the value of '$s$': $A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (20 \text{ m})^2$ $A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (400 \text{ m}^2)$ $A_{\text{hexagon}} = 3\sqrt{3} \times 200 \text{ m}^2$ $A_{\text{hexagon}} = 600\sqrt{3} \text{ m}^2$
• Using the approximation $\sqrt{3} \approx 1.732$: $A_{\text{hexagon}} \approx 600 \times 1.732 \text{ m}^2$ $A_{\text{hexagon}} \approx 1039.2 \text{ m}^2$

Fountain Area Calculation

Next, calculate the area occupied by the circular fountain.

• The fountain occupies 5% of the garden's area.
• Calculate 5% of the hexagon's area: $A_{\text{fountain}} = 5\% \text{ of } A_{\text{hexagon}}$ $A_{\text{fountain}} = 0.05 \times 1039.2 \text{ m}^2$ $A_{\text{fountain}} = 51.96 \text{ m}^2$

Remaining Area Calculation

Finally, find the remaining area by subtracting the fountain's area from the total garden area.

• Remaining Area = Total Garden Area - Fountain Area
• $A_{\text{remaining}} = A_{\text{hexagon}} - A_{\text{fountain}}$ $A_{\text{remaining}} \approx 1039.2 \text{ m}^2 - 51.96 \text{ m}^2$ $A_{\text{remaining}} \approx 987.24 \text{ m}^2$

The remaining area of the garden is approximately $987.24 \text{ m}^2$.