GmatKiss wrote:pls help
See the figure.
Since ABCDEF is a regular hexagon with perimeter 36, so each side of the hexagon = 6
So, the radius of circles O, A, B, C, D, E, and F is 6/2 = 3
Hence,
area of circle with center at O = (pi)(3)² = 9(pi)
Now, each internal angle of hexagon = (6 - 2) * 180º/6 = 120º {The Interior Angle (or Internal Angle) of a regular polygon with "n" sides can be calculated using: (n - 2) * 180° / n}
Next, Area of Sector = 1/2 * (θ * pi/180) * r² (when θ is in degrees)
So,
Area of the circles that are inside the hexagon = 6 * {1/2 * (120 * pi/180) × 3²} = 18(pi)
A regular hexagon has 6 equilateral triangles. Now, we need to find the area of hexagon.
Look at the following figure.
First we'll find the area of 1 equilateral triangle.
By Pythagoras Theorem, OD = √{s² - (s/2)²} = √(3s²/4) = s√3/2
So, area of 1 equilateral triangle = (1/2)* s * s√3/2 = s²√3/4
Therefore,
area of hexagon ABCDEF = 6s²√3/4 = 6(36)√3/4 = 54√3
So,
Area of shaded region = 54√3 - {(9(pi) + 18(pi)} = 54√3 - 27(pi)
The correct answer is E.
