Find the area of a regular hexagon with an apothem 10.4 yards long and a side 12 yards long. Round your answer to the nearest tenth. (show your work)
To find the area of a regular hexagon, we can use the formula:
Area = (3√3 * s^2) / 2
Where s is the length of a side of the hexagon.
Given that the side length (s) is 12 yards, we substitute it into the formula:
Area = (3√3 * (12)^2) / 2
Area = (3√3 * 144) / 2
Area = (432√3) / 2
Area = 216√3
Now, we need to find the apothem of the hexagon. The apothem is a line segment from the center of the hexagon to the midpoint of a side. Since the apothem is given as 10.4 yards, we can divide the hexagon into six equilateral triangles with side length 12 and apothem 10.4. Using the formula for the area of an equilateral triangle:
Area of one equilateral triangle = (1/2) * base * height
Area of one equilateral triangle = (1/2) * 12 * 10.4
Area of one equilateral triangle = 62.4 square yards
Since there are six equilateral triangles in a regular hexagon, the area of the hexagon is 6 times the area of one equilateral triangle:
Area = 6 * 62.4
Area = 374.4 square yards
Therefore, the area of the regular hexagon is approximately 374.4 square yards.