GeminiHeat wrote:
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Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)?
A. \(108-18\pi\)
B. \(54\sqrt{3}-9\pi\)
C. \(54\sqrt{3}-18\pi\)
D. \(108-27\pi\)
E. \(54\sqrt{3}-27\pi\)
Let each side of the Hexagon be \(a\)
\(6a = 36\)
\(a = 6\)
Area of Regular Hexagon \(= 6(\frac{\sqrt{3}}{4} a^2) = 54\sqrt{3}\)
Now, we can notice that each side \(a\) is equal to twice the radius \(r\) of the circles
i.e. \(6 = 2r\)
\(r = 3\)
Now, Since each interior angle of Regular Hexagon is \(120\) degrees, \((120)(6) = 720\) degrees
Therefore we will have \(2\) circles from \(720\) degrees and \(1\) circle in the centre.
Required Area = Area of Hexagon - Area of 3 Circles\(= 54\sqrt{3} - 3π(3)^2\)
\(= 54\sqrt{3} - 27π\)
Hence, option E