The length of the smaller diagonal of a regular hexagon is 3. Find: a) the length of its large diagonal; b) the area of the hexagon.

The side of the regular hexagon is two times smaller than its large diagonal. The smaller diagonal of a regular hexagon is the leg of the right triangle, the hypotenuse of which is the larger diagonal, and the side of the hexagon is the second leg. If x is the side of the hexagon, then 2x is its large diagonal and 4×2 – x2 = 9. Therefore, x = √3, and the large diagonal is 2√3. b) A regular hexagon is composed of six regular triangles with an area of 0.75√3. Therefore, the area of the hexagon is 4.5√3.

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