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hexagon ABCDEF

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by neelgandham » Fri May 18, 2012 12:57 pm
Radius of the circles A B C D E F = 3 Units.
Angle between sides of a Hexagon = 120 degrees.
Area of sector of circle A inside the Hexagon = (120/360) * pi * 3 * 3 square units.
Area of all the sectors = 6 * (120/360) * pi * 3 * 3 = 18*pi square units.
Area of the circle inside the hexagon = pi*3*3 = 9 pi
Total Area of the hexagon = (3*√3/2)*6*6 = 54√3

Area of the shaded region = Total Area of the hexagon - Area of the circle inside the hexagon - Area of all the sectors = 54√3 - 9*pi - 18*pi = 54√3 - 27*pi

Answer E
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by Anurag@Gurome » Fri May 18, 2012 10:41 pm
GmatKiss wrote:pls help
See the figure.

Image

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.

Image

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.
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