kizito2001 wrote:

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)?

My approach:

Divide the hexagon into 6 equilateral triangles:

1. Area of hexagon. = 6 * area of a triangle (As all triangles are equilateral with same side).

= \(6 * \sqrt{3} /4 * 6^2\)

= \(54\sqrt{3}\)

2. Area of orange part in the diagram:

\(pie*r^2* 120/360\)

\(pie*9* 1/3\)

\(3* pie\)

Total of 6 areas like this + the central circle = \(18 * pie + 9 * pie = 27 * pie\)

Now area of black shaded region = area point 1 - area point 2.

= \(54\sqrt{3} - 27 * pie\)

Answer E.

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