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# Regular hexagon ABCDEF has a perimeter of 36. O is the

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Regular hexagon ABCDEF has a perimeter of 36. O is the [#permalink]  09 Dec 2007, 19:36
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)?
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This is MGMAT Question =)

1) The side of a hex = 6 => r of the circle = 3.
2) Area of a hex = 6 * area of triangles with the side =6:
S = 1/2 height * base. The height divide triangle (with the sides = 6) to 2 triangles with the sides 6, 3*sqrt3 and 3 (as this is 1:sqrt3:2 triangles)
S of the triangle = 1/2 * (3*sqrt 3) * 6 = 9 * sqrt 3

So the area of hex = 6 * 9 * sqrt 3 = 54 * sqrt 3

The area of shaded region = 54 * sqrt 3 - 3 (Pi*3^2) = 54 * sqrt 3 - 27*Pi
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