Use the image to answer the question.
An illustration shows a hexagon with an inward triangle drawn from one of the sides. The triangle is marked A equals 4.
Find the area of the regular hexagon.
(1 point)
To find the area of the regular hexagon, we can split it into six equilateral triangles. Since each interior angle of a regular hexagon is 120 degrees, each angle of the equilateral triangle is 60 degrees.
We know the height of the equilateral triangle is labeled as 4.
Using the formula for the area of an equilateral triangle:
Area = (base * height) / 2
Since the base of the equilateral triangle is equal to one side of the regular hexagon, and all sides are equal in a regular hexagon, the base of the equilateral triangle is equal to the side of the hexagon.
Let x be the length of each side of the regular hexagon.
Triangle area = (x * 4) / 2 = 2x
Now we need to find the height of the equilateral triangle, which we can do by creating a right triangle:
We can see that the height of the equilateral triangle forms a 30-60-90 right triangle.
Using trigonometry:
sin(60) = opposite / hypotenuse
sin(60) = h / x
(√3 / 2) = 4 / x
x = 8 / √3
Area of the regular hexagon = 6 * (triangle area) = 6 * 2x = 6 * 2(8 / √3) = 12 * (8 / √3) = 96 / √3
Therefore, the area of the regular hexagon is 96 / √3 square units.