Answer:
E. (54\sqrt{3}-27\pi\)
Step-by-step explanation:
I draw the figure described by the statement down there so you can see easily how to solve it.
Notice 3 things:
- The length of one side of the hexagon is simply 36 divided by 6. = 6 This is beacause, by nature, every single side of the hexagon has the same length.
- The radius of each circle is half the length of one side of the hexagon. This would mean that the radius of each circle is 3.
- The outer circles have 120°/360° of its area enclose. We get this value because by definition, the hexagon can be divided into 6 regular triangles, each one with every angle equal to 60°. The size of the internal angles of the hexagon will be twice this value.
So, in summary, The area enclosed by the circles is expressed like this:
Acircles = 6 * 
* π * (r)² + π*r² = 3 * π * r² = 27 π
Now all we need is the area of the regular hexagon, which is simply:
Ahexagon = 
* p * a
Where p is the perimeter, given by the problem, and a is its apothem, the distance between the center of the hexagon and the middle of one of its sides, that is found by multiplying the length of a side of the hexagon times 
.
Ahexagon = * 36 * 6 * = 54
.
Then, the area of the shaded area is equal to:
Ashaded = Ahexagon - Acircles = 54 - 27π
