Answer:
To find the area of a regular hexagon with an apothem of 12 centimeters, we can use the formula:
Area = 1/2 * Perimeter * Apothem
In a regular hexagon, each interior angle measures 120 degrees, and the apothem is the distance from the center of the hexagon to the midpoint of a side, forming a right angle with that side.
Given that the apothem is 12 centimeters, and in a regular hexagon, the apothem is also the height of each equilateral triangle formed within the hexagon, we can find the side length of the hexagon using trigonometry:
The apothem divides the equilateral triangle into two right-angled triangles. The height (apothem) is 12 cm, and the base (side of the hexagon) is the hypotenuse of one of these right-angled triangles.
Using trigonometry (sin 60° = opposite/hypotenuse), we can find the side length:
sin 60° = 12 / side length
side length = 12 / sin 60°
side length = 12 / √3 / 2
side length = 12 * 2 / √3
side length = 24 / √3
side length = 24√3 / 3
side length = 8√3
Now, we can calculate the perimeter of the hexagon:
Perimeter = 6 * side length
Perimeter = 6 * 8√3
Perimeter = 48√3
Finally, we can find the area using the formula:
Area = 1/2 * Perimeter * Apothem
Area = 1/2 * 48√3 * 12
Area = 24 * 12√3
Area = 288√3 square centimeters
Therefore, the area of the regular hexagon with an apothem of 12 centimeters is 288√3 square centimeters.