Answer:
A hexagon has 6 internal angles, and the sum of the interior angles is 720°
If the hexagon is regular, so all the internal angles are equal, the measure of each internal angle is:
720°/6 = 120°
Ok, now let's look at the image.
We can "craft" a triangle rectangle. such that one cathetus is the line of 4ft that we can see in the image.
The hypotenuse is the line that connects the center, with the bottom right vertex.
Half of the bottom side is the other cathetus.
Because the hypotenuse divides the vertex in exactly two halves, the bottom right angle of the triangle rectangle will be 120°/2 = 60°.
So of this triangle rectangle, we know:
one angle equal to 60°
The opposite cathetus of this angle, which measures 4 ft.
Let's find the other cathetus of the rectangle:
We can use the identity:
Tan(a) = (opposite cathetus)/(adjacent cathetus)
In this case, we have:
Tan(60°) = 4ft/x
The value of x is:
x = 4ft/Tan(60°) = 4ft/√3
Because Tan(60°) = √3
And we know that the area of a triangle rectangle is the product of the length of each cathetus divided by two.
Then the area of our triangle rectangle is:
A = (4ft*4ft/√3)/2 = (8/√3) ft^2
And there are 12 of these right triangles inside the hexagon
Area = 12* (8/√3) ft^2 = (96/√3) ft^2
And we can write:
96 = √(96*96) = √9216
Then:
Area = √(9216/3) ft^2 = √(3072) ft^2