4. 7. Use the apothem and perimeter to find the area of a regular hexagon with side length 12 feet.

To find the area of a regular hexagon, you can use the formula \(\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\). First, calculate the perimeter: with a side length of 12 feet, the perimeter is \(6 \times 12 = 72\) feet. Next, the apothem, which can be found using the formula \(a = \frac{s}{2 \tan(180/n)}\) where \(s\) is the side length and \(n\) the number of sides, gives us approximately 10.39 feet. Thus, the area calculates roughly to \( \frac{1}{2} \times 72 \times 10.39 \approx 374.04\) square feet! Speaking of geometry, did you know that the concept of polygons dates all the way back to ancient Greece? Mathematicians like Euclid studied these shapes extensively, laying the groundwork for modern geometry. Hexagons, in particular, are fascinating due to their efficiency in tiling and structural integrity—think of honeycombs! They use minimal material while maximizing storage space, an inspiring example of nature’s genius in geometry.