Solve the problem: Calculate the area of a regular hexagon with a side length of 8 cm.

Problem: Area of a Regular Hexagon

We need to calculate the area of a regular hexagon with a side length of 8 cm.

A regular hexagon can be divided into six equilateral triangles. The area of an equilateral triangle with side length 'a' is given by the formula:

$$ A_{\text{triangle}} = \frac{\sqrt{3}}{4} a^2 $$

The area of a regular hexagon is 6 times the area of one of these equilateral triangles:

$$ A_{\text{hexagon}} = 6 \times A_{\text{triangle}} = 6 \times \frac{\sqrt{3}}{4} a^2 = \frac{3\sqrt{3}}{2} a^2 $$

Given the side length $$a = 8$$ cm:

$$ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (8 \text{ cm})^2 $$$$ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} (64 \text{ cm}^2) $$$$ A_{\text{hexagon}} = 3\sqrt{3} \times 32 \text{ cm}^2 $$$$ A_{\text{hexagon}} = 96\sqrt{3} \text{ cm}^2 $$

To get a numerical value, we can approximate $$\sqrt{3} \approx 1.732$$:

$$ A_{\text{hexagon}} \approx 96 \times 1.732 \text{ cm}^2 $$$$ A_{\text{hexagon}} \approx 166.272 \text{ cm}^2 $$

Answer: The area of the regular hexagon is $$96\sqrt{3}$$ cm² (approximately 166.27 cm²).